Here it is:
The Mysteries seem to originate in periods of social flux and uncertainty that undermined the normal community oriented religion.
As the article on the Mysteries and the Hellenistic Age which I posted below, and Nietzsche argues (in The Twilight of the Idols), the community oriented religious cults of state, family and home focused on this world. Proper performance would perpetuate and strengthen the community, which was one's ultimate legacy and immortality. The Gods, especially the patron gods of cities and nations reflected and promised these aspirations. Obedience to YHVH, attention to the Pax Deorum or Eusebia would lead to children, and national wealth and power in this life. Disobedience would lead to disaster.
This attitude is rampant throughout the OT, when Judaism was still a tribal religion with a tribal, God, and its found throughout Hellenic and Roman society. Sokrates was put to death for fear he would anger the Gods with his new "things spiritual", Alcibiades was hunted down for defacing the Hermae and threatening good relations with the Gods. No doubt the disaster at Syracuse caused pious Athenians to think the Gods were punishing that impiety. Conseravative Romans warned that the abandonment of the Gods by the Christian Augusti had led to the Gods abandoning the Imperium...and one must wonder i they were not right. When Pulcher drowned the Sacred Chickens, no one was suprised when his fleet went down with him in a storm. When Israel abandoned YHVH, the Assyrians and Neo-Babylonians ravaged the land.
However, when one is pious, and yet still disaster falls upon you...then people begin to hope for a future life, where the good get rewarded and the evil are punished. In the increasingly footloose and unsettled Hellenistic Age, when distant rulers ruled, despite what one did in one's Polis, or nation...then we see myths arise about such judgment.
Plato, writing in the aftermath of the stinging defeat of Athens by Sparta writes in the Republic of the Myth of Er. Here we see wrongdowers punished, and the good rewarded with time in Tartaros and Elysium respectively prior to their rebirth. In the Hellenistic Age too, we see discussions of the judgment of the dead by Minos, Rhadymanthus and Sarpedon deciding where to send the deceased. In Judaism, the prospect of Hell and Torment is borrowed by the Jews from the Zoroastrians. The Zoroastrians themselves, once had a typical "House of Clay" for the dead...with the Rich and Powerful and Pious going to a better realm. Zoroaster, in his rebellion against the wars and conquests of the Indo-Iranian followers of Indra, turned to the ethical uprightness of Apam Napat (Varuna or Ahura Mazda) and developed the idea of judgement where the virtuous get rewarded.
In that Axial Age, Xenophanes and Plato and Pythagoras turn against the amoral mythology of Hellas, and develop ideas of the Gods as Moral Agents and Guardians. In their ethical reform of Hellenic religion, the Gods can do no wrong and evil. The poets who say otherwise are liars and should be banned or censored. The line in Homer where Zeus has at his hand two jars, one of benefits, and one of evils, which he distributes at his pleasure draws particular criticism from Platon in The Republic. While the comforting lines of Hesiod, from another troubled time, the Dark Ages of Hellas, about how Justice resides with Zeus, and his invisible watchers (shades again of the watchers of Varuna) move across the land to report on the unjust...and how Zeus punishes them, no doubt drew approval from the broad shouldered Wrestler turned Philosopher.
Meanwhile, in Judaea, the Prophets make a similar Ethical Revolution in Yahwism, with YHVH becoming the stern, upright , moral, judge of the world. Where before he is a typical mythological God, he takes on a judge aspect, similar to Varuna/Ahura Mazda, or the purified form of Zeus seen in Chryssipus' Hymn to Zeus or Plato. Some argue that the Persians themselves imposed this view on the Jewish Scriptures after the Exile, though if the traditional chronology is true, this trend predates the Exile, though post-Exilic times and experiences (and Persian influence) may have accelerated it.
In India, Mahavira and Siddartha Guatama undertake moral reforms of their own of the Vedic religion present there. In old Buddhist texts, we see Indra become a Buddha who tries to encourage Siddartha to become a Buddha himself. Indra rules because he is a Buddha, the greatest so far, but Siddartha would be greater still. Moral reforms against the purification laws, the extreme asceticism, and evolving Caste system lead to Buddhism. In response, Hinduism forms, taking ideas and influences from Buddhism to transform Vedism into Hinduism as we know it today, also with a moral reform element.
We can also note how the Mysteries seem to loom out of, or appear in times of disarray and disorder and social flux. From the Dark Ages come the Kebeiroi, the Korybantes, Kuretes, and Eleusis. The Telesterion of Eleusis was built in that early period, and expanded in the Archaic Age...on the ruins of a Mycenean Megaron (the Palace of the King of Eleusis where key elements of the Persephone myth and Raging Demeter's travels occur...indeed where the Mysteries are revealed!). The Dark Ages were a time of flux, of war, poverty, collapse of civilization, social unrest (the Monarchies fall to the Oligarchies and Aristocracies across Hellas). Colonies were sent out, often with unwilling, conscripted colonists. Prominent citizens may find themselves on a distant shore founding a new city in virtual exile, while poor folk may find new power in their new land.
Is it any surprise that so many Mysteries arise in this Period. Oftentimes a new deity comes to Hellas, bringing a Mystery. Such as Dionysos. He is not found in the Linear B tablets. (Neither is his antithesis, with whom He shares Delphi, Apollo, apparently an Anatolian Deity). Legend says he came from Thrace, where many colonies sprouted in the Dark Ages. Samothrace, and Lesbos on the borders of the known world, near the new colonies are centers of other Mystery religions. A result of the new and old mixing in the social flux of the age?
In the Hellenistic Age, other Mysteries appear, quite suddenly. Mithraism develops in Pergamum and spreads to Rome. Hellenes living in Egypt fuse Dionysus, the Apis Bull and Osiris to make Serapis the Savior and his devoted wife Isis...the Serapean or Isean Mysteries. From Syria came other Mysteries, and from Phrygia came Kybele with her Gallae.
What do we know of the Mysteries? Not much, though perhaps more than winegodeatsyou implies. In Christian times, some Patristic Fathers were initiates and in their polemics reveal details of the Mysteries. One of the Playwrights of Classical Athens was an initiate, and revealed enough details in a play that the audience, consisting mostly of other initiates went after him with intention to have a lynching. Murals in Pompeii reveal other details. Apuleius revealed something of the Isean Mysteries.
Importantly, the funerary remains of inscriptions of the Dionysian Mysteries reveal a promise of an afterlife in Elysium or amongst the Gods, a much better place than the purgatory like Sheol or Hades. Salvation in the next life seems important...as can be seen in Gnosticism, Christianity, Essenism, and other Jewish Mysteries that evolved after despite the best efforts of the Hassid, the Gentiles still win despite YHVH's promises...and a reinterpretation of that promise into the Next World.
So we know a bit about them. But the key item, the "Apostolic succession" of initiation has been broken. There are no direct descendants of the Mysteries today...though there are rumors, unsubstantiated, that Mithraism may have persisted in certain Sodalities (many of which were made up of initiates) practicing a certain craft..which, after mixing with Christianity flourishes today as Freemasonry. Whether this is true or not, I know not. Certainly the focus on ethics, the male-only nature of the rites, and the contrast of light v. dark in the tiles of the floor, and in the symbolism, and the tomb in which one is brought into the light suggest certain Zoroastrian/Persian ideas and the cavelike (tomblike?) Mithraea of Rome itself. Interestingly, the aspect of Mithraism which ursus77likes...the presence of successful and prominent citizens continues to this day in Masonry.
In times of trouble, or in normal life, the poorest would be the most vulnerable and the ones most in need of succor in the next life. Hence is it any surprise that the Mysteries attracted usually the very well off seeking a thrill much as modern Cults and Mysteries attract the Wealthy, the Diletants and others seeking enlightenment and comfort. But the consolation of the Mysteries would also attract the poor, the slaves, the women...the mostly dispossessed of the society. It grants them important titles such a priest, initiate a sense of being better than their uninitiated social betters.
Once again, with penetrating insight, Nietzsche isolates this element of ressentiment in the Mysteries and the classes most attracted to them. He explains their spread and popularity...and their eventual eclipse by the most successful Mystery Religion of all...no, not Mithraism...but Christianity...which fuses Jewish and Pagan Mysteries, combines with a strong element of philosophy (another consolation which I will discuss later).
Forgive any possible impiety in what I say, I merely offer an explanation. In defense of the Mysteries, a feeling of belonging to the group of the Elect, a desire for consolation about one's fate in the next life, and a desire for a Theodicy are all valid needs that must be fulfilled. These are needs the Mysteries fulfilled quite well. These are needs others feel, and which I feel, and which is one reason why I track the efforts by some to recreate (or more accurately create) Mysteries today.
Now, as I explain, above, we know something of the Mysteries. But I agree with winegodeatsyou that we cannot reconstruct specific mysteries. We can only study the elements that went into them. Rather than try to recreate the Mystries of Eleusis, for instance, despite the massacre of the Hierophants when the Arian Visigoths sacked the Telesterion...one could try to create, say the Mysteries of Richmond, or Boston, or what not. We might be able to recreate the Mithraic Mysteries...and I'm a member of Mithraeum, a group slowly working towards that. However, at some point some divine revelation will be needed, akin to the revelation to Orpheus, or from Demeter to the Queen of Eleusis, and so forth. This will, of necessity involve a degree of UPG, both on the part of the Founder and of the future initiates. I.e., a leap of faith.
The question of the viability of creating new Mysteries will ultimately rest on the ability of others of us to make that Leap of Faith, and accept the validity of the revelation behind the Mystery and the investiture of divine authority on the initator. In essence, the Founder or Founders would have to be initiated by the Gods...and others would have to accept that initation by faith.
Now some of us will have "cult" alarm bells ringing full tilt, but in essence, that is what is needed. Unlike the only surviving continuous Mystery of the Ancient World, Christianity, the Mysteries were non-exclusive, and apparently non-dogmatic (aside from oaths to keep secrets and some beliefs about the afterlife, they had no credo to believe, though some had behavioral codes and restrictions). Hence they lived peacefully side by side with the other cults, the State cults, the household cults, and the philosophical sects and schools. Pauline (Catholic/Orthodox & successors i.e., Protestant) Christianity did not, and killed off the competing Mysteries and cults.
However, so long as the new Mysteries remain as open, tolerant and unorthodox/non-credo religions like their (often quite syncretic) ancient predecessors, I see no reason to fear them. No one will force you to be initiated...just as no one was forced to be initiated in the ancient Mystery Cults.
You see, one thing I find wonderful about the religions of the ancient world (they're a "religion" only in the sense that the various sects and religions we call Hinduism are "a" religion...by means of an umbrella term at best)...is the variety. If one is of a traditional bent, or successful and self-satisfied, the household and state cults sufficed. If one were intellectually curious, or had a penchant for mysticism, the Philosophical schools could satisfy you along side the above cults...and for the serious mystic, or for someone worried about the afterlife, the Mysteries (one or more or all) could satisfy you.
You could belong to all, or to some as need may be. Someone like Cato the Elder, or Cicero's mouthpiece Cotta...would be perfectly happy taking part only in the state and family cultus. That would be folks like ursus77. Then there are the mystics who wish to take part in both, like say, kyrene. Then there are those who like Philosophy like myself. I am, personally interested in all of them...though the Mysteries appeal more as an insurance policy, perhaps.
Its a bit like the situation in Japan. One can take part in the Shinto religion, a religion much like that of Dodekatheism or Religio...and back that social control element with the teachings of the Philosopher, the Great Master K'ung (K'ung-Fu-Tzu or Confucius), if one desires. One could also seek enlightenment in the philosophy of Siddartha Gautama or Master Lao (Lao-Tzu). To be sure of salvation, one could worship in Pure Land Buddhism and be sure of salvation. It used to be said, born and married Shinto, buried Buddhist. Now, of course, its born Shinto, married Christian, and buried Buddhist.
And I'll note that Confucian Mandarins were convinced that Buddhism would undermine State and Family. The founder, they pointed out, abandoned his duties as ruler, and abandoned his wife and infant son. The monks contributed not to society or the State, subsisting off alms. So they prosecuted Buddhism. Only after a modified form of Buddhism, not alien to family and State formed as Mahayana Buddhism did it find acceptance in China. (Vedics and Hindus also were horrified at how Buddhists abandoned the 4 stages of life, and dharma and dharman until Buddhism had its own Constantine moment with Asoka Maurya). Taoism, that esoteric, anarchic philosophy also conflicted with Confucianism until it, in a sense returned to its roots in traditional Chinese religion and became Religious Taoism with the hierarchies of Gods and Emperor that it really lacked prosecution.
It is in the modern East that we see something like the flexibility of options that the Ancient Classical, Hellenistic and Roman world had to meet the needs of the people. Much like Classical religion(s), Hinduism today has devotionalistic cults (and ancient writers condemned the deisidaimonia -- supersitition or excessive piety) of the poor who dressed statues of the Gods, did their hair, and washed them constantly, shades of excessive Bhakti Yoga), philosophical schools and speculations (jñana yoga) such as Vedanta and Smrti, traditonal duty-based religion (Karma-Yoga) and so forth...they even have their own salvific cults reminiscient of the Mysteries (Caitana Vaisnava).
If it were not for Constantine, and Christianity's exclusive, orthodox orientation...it would have simply been and stayed one Mystery Cult among others, well integrated into the framework of Romano-Hellenistic society. Indeed, if you read Wiliken's The Christians as the Romans Saw Them you'd see that Porphyry and other apologists strived hard to fit Christianity into that mold. A mold well exampled by Augustus Alexander Severus' private chapel, with statues of Jesus, Abraham, Orpheus and other Sages on it, even while he continued as an orthoprax and otherwise conventional Pontifex Maximus.
So all in all, I see no threat in the attempt to create a New Mystery(ies) today, nor see why it wouldn't work (with the proper leap of faith). Being a modern creation, and while modeled as closely as possible on ancient Mysteries, there is no reason why they could not be made to conform to modern epistemology.
Now as to suggestions as to the motives of some of those who seek to found such modern Mysteries, I cannot agree or disagree with ursus77, in that I am not privy to their motives. Most of the ones I've met online seem quite sincere in their motivations. There is no reason to disbelieve them...though it is possible that some are seeking status, perhaps from a Nietzschean ressentiment motive. I simply cannot say.