Old English beste, reduced by assimilation of -t- from earlier Old English betst "of the highest quality or standing, first, in the best manner." This originally was the superlative of bōt "remedy, reparation" (Middle English bote "advantage, help, profit"), a word now surviving in its simple form only in the expression to boot (see boot (n.2)). Its comparative, better, and superlative, best, have been transferred to good (and in some cases well).
Old English bōt is from Proto-Germanic root *bat-, with comparative *batizon and superlative *batistaz. The superlative form is the source also of Old Frisian, Old Saxon, Middle Dutch best, Old High German bezzist, German best, Old Norse beztr, Gothic batists. Also in Old English as an adverb, "in the most excellent manner."
The best-laid schemes o' mice an' men
Gang aft agley,
An' lea'e us nought but grief an' pain,
For promis'd joy!
[Burns, from "To a Mouse, on Turning Her Up in Her Nest With the Plough, November, 1785"]
From late Old English as "of greatest advantage, most suitable." Best-seller as short for "best-selling book" is from 1889, apparently originally in the publishing trade; best friend was in Chaucer (late 14c.). Best girl is first attested 1881, American English; best man is 1814, originally Scottish, replacing groomsman.
Old English habban "to own, possess; be subject to, experience," from Proto-Germanic *habejanan (source also of Old Norse hafa, Old Saxon hebbjan, Old Frisian habba, German haben, Gothic haban "to have"), from PIE root *kap- "to grasp." Not related to Latin habere, despite similarity in form and sense; the Latin cognate is capere "seize.
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 (as in 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 and readiness to perform anywhere.
'There are but two families in the world, as my grandmother used to say; "the Have's and the Have-not's," and she stuck to the former; and now-a-days, master Don Quixote, people are more inclined to feel the pulse of Have than of Know.' ["Don Quixote de la Mancha," transl. Charles Jarvis, London, 1742]