1590s, "covering into which both hands may be thrust to keep them warm," from Dutch mof "a muff," shortened from Middle Dutch moffel "mitten, muff," from French moufle "mitten," from Old French mofle "thick glove, large mitten, handcuffs" (9c.), from Medieval Latin muffula "a muff," a word of unknown origin.
The muff was introduced into France toward the close of the sixteenth century, and soon after into England. It was used by both men and women, and in the seventeenth century was often an essential part of the dress of a man of fashion; but it is now exclusively an article of female apparel. [Century Dictionary]
Meaning "vulva and pubic hair" is from 1690s; muff-diver "one who performs cunnilingus" is from 1935.
"to bungle, perform clumsily or badly," by 1840, said to be from pugilism slang, probably related to muff (n.) "awkward person, simpleton" (1837), which is perhaps from muff (n.) on notion of someone clumsy because his hands are in a muff. Related: Muffed; muffing. The noun meaning "anything done in clumsy or bungling fashion" is by 1871.
early 15c., "to cover or wrap (something) to conceal or protect," perhaps from Old French moufle "thick glove, mitten;" see muff (n.). Compare Old French enmoufle "wrapped up;" Middle French mofler "to stuff." The meaning "wrap something up to deaden sound" is recorded by 1761. Related: Muffled; muffling. Muffled oars have mats or canvas about their shafts to prevent noise from contact with the oarlocks while rowing.
in cookery, 1946, from Italian manicotti, said to mean literally "hand-warmers, muff," from Latin manicae "long sleeves of a tunic, gloves; armlets, gauntlets; handcuffs" (see manacle (n.)).
"ponder, turn over in one's mind," 1873, perhaps from a figurative use of mull (v.) "grind to powder" (which survived into 19c. in dialect), from Middle English mullyn, mollen "grind to powder, soften by pulverizing," also "to fondle or pet" (late 14c.), from Old French moillier and directly from Medieval Latin molliare,mulliare, from Latin molere "to grind," from PIE root *mele- "to crush, grind."
Of uncertain connection to the mull (v.) defined in Webster's (1879) as "to work steadily without accomplishing much," and the earlier identical word in athletics meaning "to botch, muff" (1862). Related: Mulled; mulling.