late 13c., "a gift, a reward, a favor bestowed freely;" c. 1300, "goodness, virtue; beauty; ; excellence; knightly prowess, strength, valor, chivalry," early 14c., "a helpful act, an act of generosity, a good deed," also "liberality in giving, generosity, munificence," from Anglo-French bountee, Old French bonte "goodness" (12c., Modern French bonté), from Latin bonitatem (nominative bonitas) "goodness," from bonus "good" (see bonus).
Sense of "gift bestowed by a sovereign or the state" led to extended senses of "premium or gratuity to a military recruit" (1702) and "reward for killing or taking a criminal or enemy" (1764) or dangerous animal (1847). Bounty-jumper, "one who enlists in the military, collects the bounty, and flees without reporting for duty" is from the American Civil War (by 1864). Bounty-hunter is from 1893, American English, originally in reference to wild animals.
I do ... promise, that there shall be paid ... the following several and respective premiums and Bounties for the prisoners and Scalps of the Enemy Indians that shall be taken or killed .... ["Papers of the Governor of Pennsylvania," 1764]