"one who performs in a mumming, actor in a dumb show," early 15c., probably a fusion of Old French momeur "mummer" (from Old French momer "mask oneself," from momon "mask") and Middle English mommen "to mutter, be silent," which is the source of mum (interjection). "[S]pecifically, in England, one of a company of persons who go from house to house at Christmas performing a kind of play, the subject being generally St. George and the Dragon, with sundry whimsical adjuncts" [Century Dictionary].
1520s, "a show or performance of mumming," from Old French mommerie, from momer "to mask oneself" (see mummer). Transferred sense of "ridiculous ceremony or ritual" is from 1540s.
derisive word for an old man, 1885, according to OED a variant of obsolete Cockney guiser "mummer, one wearing a mask or costume as part of a performance" (late 15c.; see guise). If so, the original notion was "one who went about in disguise," hence "odd man," hence "old man" (it still commonly is qualified by old).