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 In English it superseded bet in the adverbial sense by 1600.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.
Old English habban "to own, possess; be subject to, experience," from Proto-Germanic *haben- (cf. Old Norse hafa, Old Saxon hebbjan, Old Frisian habba, German haben, Gothic haban "to have"), from PIE *kap- "to grasp" (see capable). Not related to Latin habere, despite similarity in form and sense; the Latin cognate is capere "seize." Old English second person singular present hæfst, third person singular present hæfð became Middle English hast, hath, while Old English -bb- became -v- in have. The past participle had developed from Old English gehæfd.
Sense of "possess, have at one's disposal" (I have a book) is a shift from older languages, where the thing possessed was made the subject and the possessor took the dative case (e.g. Latin est mihi liber "I have a book," literally "there is to me a book"). Used as an auxiliary in Old English, too (especially to form present perfect tense); the word has taken on more functions over time; Modern English he had better would have been Old English him (dative) wære betere. To have to for "must" (1570s) is from sense of "possess as a duty or thing to be done" (Old English). Phrase have a nice day as a salutation after a commercial transaction attested by 1970, American English. Phrase have (noun), will (verb) is from 1954, originally from comedian Bob Hope, in the form Have tux, will travel; Hope described this as typical of vaudevillians' ads in "Variety," indicating a willingness to perform anywhere, any time.