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Mystery derives from the ancient Greek mysterion, referencing mystery religions such as the Eleusinian Mysteries (Oxford Dictionaries).  The secrets into which participants were initiated could be aporrheton, meaning “forbidden”, and/or arrheton, meaning “unutterable, unspeakable, ineffable” (see Burkert).

From a naturalistic and scientific perspective, the aporrheton aspect no longer seems appropriate, but there would still seem to be a powerful place for the arrheton.  For example, Ursula Goodenough describes an experience of mystery before the awesome cosmos in her book The Sacred Depths of Nature, excerpted here.

Many fear that introducing scientific explanation into traditionally religious or spiritual realms of experience will destroy the mystery.  However, upon consideration, that does not seem to be the case.  For all we discover about how the universe works, it still remains a breathtaking marvel that the universe is as it is.  This is expressed, for example, in Richard Dawkins’ The Magic of Reality.  To the extent that reality remains ineffable in its fullness no matter how much we say or attempt to say about it, arrheton endures.

This view is affirmed in Tom Clark’s essay Spirituality Without Faith:

When we confront the startling fact that existence isn’t subsumable under any overarching interpretation, but simply is, we are left with an irreducible mystery about why we are here, or exist at all; and mystery serves at least as well as purpose to inspire spiritual experience. Unable not to ask questions about ultimate purpose and meaning, but rebuffed by the logic which shows such questions unanswerable, we are caught in a cosmic perplexity, a state of profound existential astonishment. The realization that existence inevitably outruns our attempts to assign meaning and purpose can have the impact of a true revelation, stunning the discursive mind in the manner of a Zen koan. Like a koan or other practices in which thinking confronts its own limitations, such a cognitive impasse can serve as the gateway to the direct, non-discursive experience that the present is sufficient unto itself. After all, there is no place to get to, no goal toward which Being is moving.

A different use of “mystery” is invoked by Noam Chomsky when he distinguishes between puzzles, which are solvable in principle, and mysteries, which are by their very nature unsolvable.  Whether mysteries exist in this sense is controversial at present.  New mysterianism is one current approach to the hard problem of consciousness.

An idea similar to Chomsky’s is put forward by economist Kaplan, who distinguishes between problems, which may be overcome, and predicaments, which cannot be overcome but only confronted.  The latter seems very close to Brendan Myers’ notion of an Immensity, though Myers himself does not use Kaplan’s terms.  Death and the passage of time, for example, are existential realities with which one may only come to terms (barring future development of fantastic technologies, at least), and in coming to terms one’s identity and moral character are called into question (see “Immensity” below).

The relationships between these three concepts – Chomsky’s mystery, Kaplan’s predicament, and Myers’ Immensity – suggest there may be a moral quality to the sense of mystery before the cosmos: it demands a response that questions who you are and how you will live your life.  As in the ancient mysteries, there is a sense in which one is changed, initiated if you will, by the experience.  Thereafter, one lives in a different and expanded universe.

See also “Mystical” and “Immensity.”

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