"be silent," 1560s, from a verb mum (Middle English mommen) "make silent" (c. 1400); "be silent" (mid-15c.), from mum, mom (late 14c.), "an inarticulate closed-mouth sound" indicative of unwillingness or inability to speak, probably imitative. As an adjective meaning "secret" or "silent" from 1520s. Phrase mum's the word is recorded by 1704.
late 14c., "very brief portion of time, instant," in moment of time, from Old French moment (12c.) "moment, minute; importance, weight, value" and directly from Latin momentum "movement, motion; moving power; alteration, change;" also "short time, instant" (also source of Spanish, Italian momento), contraction of *movimentum, from movere "to move" (from PIE root *meue- "to push away").
Some (but not OED) explain the sense evolution of the Latin word by notion of a particle so small it would just "move" the pointer of a scale, which led to the transferred sense of "minute time division."
In careful use, a moment has duration, an instant does not. The sense of "notable importance, 'weight,' value, consequence" is attested in English from 1520s. Meaning "opportunity" (as in seize the moment) is from 1781.
In for the moment "temporarily, so far as the near future is concerned" (1883) it means "the present time." Phrase never a dull moment is attested by 1885 (Jerome K. Jerome, "On the Stage - and Off"). Phrase moment of truth first recorded 1932 in Hemingway's "Death in the Afternoon," from Spanish el momento de la verdad, the final sword-thrust in a bull-fight.
"lasting but a moment, of short duration," mid-15c., momentare, from Late Latin momentarius "of brief duration," from Latin momentum "a short time, an instant" (see moment).
momentarily, momently. The first means for a moment (he was momentarily abashed), the second from moment to moment or every moment (am momently expecting a wire from him). The differentiation is well worth more faithful observance than it gets; & the substitution of either, which sometimes occurs, for instantly or immediately or at once is foolish novelty-hunting. [Fowler, 1926]
"buffoon, fool, stupid person," 1550s, from Old French mome "a mask. Related" Momish. The adjective introduced by "Lewis Carroll" is an unrelated nonsense word.
"humorously disagreeable person," 1560s, from Latinized form of Greek Mōmos, name of the god of ridicule and sarcasm (literally "blame, ridicule, disgrace," a word of unknown origin); also used in English as personification of fault-finding and captious criticism.