Greek myein, to close (eyes, lips): the myesis was the closing, initiation, and the mystes the closed one. Eleusis, Samothrace, the Dionysiac and Orphic rites ran for over a thousand years on this architecture: secret ceremonies, vows of silence, and a secret so well kept we still do not know what was shown.
The modern inversion: mystery now means an unsolved problem, a whodunit. The word flipped from forbidden-to-tell to unable-to-know, secrecy reassigned from the initiate to the universe.
Myein's root *mu-, to murmur, hum, close the lips, also gave mute, mum, mumble, and by folk extension, the mumbo in mumbo jumbo. The mystery religions and the speechless sound share a root: holiness as the thing said with the mouth shut.
If a mystery is a thing you may not tell, what is a religion you may not Google?