Wordorigins.org

Handicap comes from an old method of setting odds in a horse race or other contest. 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, or 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 bettor. If both agreed on either forfeiting or going ahead with the wager, then the umpire kept the money as payment. The word is a shortening of hand in the cap.

How did a word for a taxi also become a term meaning overused and worn out?

Hackney comes from the Old French haquenée, meaning a gentle, riding horse, an ambling horse. It was adopted into English in the 14th century meaning a horse of middle size or fair quality. From The Romance of Sir Beues of Hamtoun, a 14th century poem:

Ac nim a ligter hakenai & lef her the swerd Morgelai.
(But take a lighter hackney & leave here the sword Morgelai.)

Very early on, by 1393 at the latest, the word had also acquired the meaning of a horse for hire. From William Langland’s Piers Plowman (C Text) of that year:

Ac hakeneyes hadde thei none. bote hakeneyes to hyre.
(But hackneys had they none, hackneys for hire to boot.)

The metaphor behind this modern slang term for a computer enthusiast, particularly one who breaks into other computer systems, is not certain, but it is probably one of continually hacking or chopping away at something until it finally gives way.

The noun hack, meaning an attempt or a try at something, is nearly 200 years old. It dates to as early as 1836 when it appears in “Davy Crockett’s” Exploits and Adventures in Texas (this book was alleged to be based on Crockett’s diary, but is a fraud; who wrote it, however, doesn’t matter when it comes to lexical evidence):

Better take a hack by way of trying your luck at guessing.¹