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.