Old English bettra, earlier betera, from Proto-Germanic *batizo-, from PIE *bhad- "good;" see best. Comparative adjective of good in the older Germanic languages (cf. Old Frisian betera, Old Saxon betiro, Old Norse betr, Danish bedre, Old High German bezziro, German besser, Gothic batiza). In English it superseded bet in the adverbial sense by 1600. Better half "wife" is first attested 1570s.
late 12c., "that which is better," from better (adj.). Specific meaning "one's superior" is from early 14c. To get the better of (someone) is from 1650s, from better in a sense of "superiority, mastery," which is recorded from mid-15c.
Old English *beterian "improve, amend, make better," from Proto-Germanic *batizojan (cf. Old Frisian beteria, Dutch beteren, Old Norse betra, Old High German baziron, German bessern), from *batiz- (see better (adj.)). Related: Bettered; bettering.
To surpass or outbid; raise the standard: That wasn't a bad offer, but I'll go you one better
[1845+ Gamblers; fr the raising of bets in poker]