Abarim Publications' online Biblical Greek Dictionary
The verb μυω (muo) and its future form μυσω (muso) predominantly means to shut (of eyes or the mouth), but with the secondary meaning of to cover, to hide or to be just under the surface.
It is the root of the word μυς (mys), meaning both muscle (hidden just under the skin) and mouse (hidden just under the floor — our English word "muscle" is the diminutive form of the Latin word for mouse). The link exists even in Semitic languages: the Arabic word for muscle is 'adalah and the word for mouse is 'adal.
Perhaps via the mess mice make, or the tendency of people to hide their dirt, or perhaps completely unrelated are the noun μυσος (musos), meaning uncleanness or defilement, and the adjective μυσαρος (musaros), meaning foul, dirty, and thus loathsome and abominable.
On the other hand, our verb μυω (muo), meaning to hide or cover, may also have spawned — or perhaps merely lubricated the formation of — the familiar word μουσα (mousa) or Muse (hence our words "music" and "museum"), which is commonly taken from the same proto-Indo-European root as the English word "mind". The adjective μουσικος (mousikos), whether derived from the above or merely inspired by it or having nothing to do with it, means devoted to the muses: artistic or musical. This word occurs in the Bible only in Revelation 18:22.
The root verb μυω (muo) is not used in the Bible, but from it comes the verb μυεω (mueo), which means to initiate into the mysteries (the hidden things) or to be introduced to things not known before or not commonly known. This verb occurs in the Bible only in Philippians 4:12.
From the previous verb derives the noun μυστης (mustes; not used in the Bible), which denotes a person initiated into the mysteries, and from the latter in turn derives the noun μυστηριον (musterion), which denotes a mystery, secret or bit of knowledge. But note that in antiquity, the wizard (which means "wise-man") was not a magician or entertainer but the doctor and chief engineer of a tribe or village. His knowledge may have seemed magic to the uninitiated but it was not supernatural in any way, but rather based on what we would call today scientific and technological skills.
Even by usually sound commentators on the Bible, the noun μυστηριον (musterion), or "mystery", is commonly rendered a far too great hocus-pocus factor. A "mystery" is not an enigma, but something hidden just under the surface, something that can be pried out from under observable reality just like a mouse can be pried out from under its cover. A mystery is a "mousery"; a nest of living mice cuddled under a flimsy cover, or a "musery"; a fountain of inspiration suddenly bursting from the earth.
The Greek idea of the "mystery" quite literally refers to a "discovery" or to something "to-be-discovered". It is the same thing as a modern scientific theory, and has nothing to do with classified information or unsolvable puzzles, and everything with the truth behind observable reality.
Neither the "mystery of iniquity" (2 Thessalonians 2:7), nor the "mystery of the faith" (1 Timothy 3:9), nor even the "mystery of Christ" (Ephesians 3:4) are anything other than descriptions of the natural causes of visible effects (Hebrews 11:1), which can explained by someone who gets it, and be made understandable by means of a verbal explanation.
Hence Paul speaks of being informed of a mystery (Romans 11:25), of telling a mystery (1 Corinthians 15:51), of being stewards of a mystery (1 Corinthians 4:1) and even "full assurance of understanding, resulting in a true knowledge of God's mystery" (Colossians 2:2).