in wagering, "equalizing allowance to a weaker side or player by a stronger, advantage conceded by one of the parties in proportion to the assumed chances in his favor," 1590s, found first in Shakespeare ("2 Henry IV," 1597), probably from the word's earlier sense of "condition of inequality, difference, amount by which one thing exceeds or falls short of another" (1540s), from odd (q.v.), though the exact sense evolution is uncertain. Odds was used for "unequal things, matters, or conditions" from c. 1500, and the later senses may have evolved generally from this earlier notion of "things that don't come out even."
Until 19c. treated as a singular, though obviously a plural (compare news). General sense of "chance or balance of probability in favor of something happening" is by 1580s. Sense of "disagreement, variance, strife" (1580s) is the notion in at odds "at controversy or quarrel, unable to agree." Odds-on "on which the odds are laid" is by 1890.
Chinese dish, 1885, American English, from Chinese (Cantonese dialect) tsap sui "odds and ends, miscellaneous bits." A Cantonese dish brought to the U.S. West Coast by Chinese immigrants.
1690s, "work composed of ill-sorted parts clumsily put together;" 1720s (though perhaps the older sense) "work composed of pieces of various colors or figures sewed together;" from patch (n.1) + work (n.). As an adjective, "made up of odds and ends," from 1713.
1650s, from hand in cap, a game whereby two bettors would engage a neutral umpire to determine the odds in an unequal contest. The bettors would put their hands holding forfeit money into a hat or cap. The umpire would announce the odds and the bettors would withdraw their hands — hands full meaning that they accepted the odds and the bet was on, hands empty meaning they did not accept the bet and were willing to forfeit the money. If one forfeited, then the money went to the other. If both agreed either on forfeiting or going ahead with the wager, then the umpire kept the money as payment. The custom, though not the name, is attested from 14c. ("Piers Plowman").
Reference to horse racing is 1754 (Handy-Cap Match), where the umpire decrees the superior horse should carry extra weight as a "handicap;" this led to sense of "encumbrance, disability" first recorded 1890. The main modern sense, "a mental or physical disability," is the last to develop, early 20c.
bite-sized square pasta packet containing various food, a classic Italian dish, 1610s, from Middle English raffyolys, also rafyols (late 14c.). The word probably was re-borrowed several times, most recently in 1841, from Italian ravioli, a dialectal plural of raviolo, a diminutive of an unidentified noun, perhaps of rava "turnip" (presuming they were among the original ingredients) or Genoese dialect rabiole "leftovers, odds and ends."
dish of chopped entrails, c. 1400, now chiefly Scottish, but it was common throughout England to c. 1700, of uncertain origin. Perhaps from Old French hacheiz "minced meat," from agace "magpie," on analogy of the odds and ends the bird collects. The other theory [Klein, Watkins, The Middle English Compendium] traces it to Old English haggen "to chop," or directly from Old Norse höggva "to hew, cut, strike, smite" (see hack (v.1)).