"manner of behaving (whether good or bad), conduct, manners," late 15c., essentially from behave, but with ending from Middle English havour "possession," a word altered (by influence of have) from aver, noun use of Old French verb aveir "to have."
early 15c., reflexive, "conduct or comport" (oneself, in a specified manner), from be- intensive prefix + have in sense of "to have or bear (oneself) in a particular way, comport" (compare German sich behaben, French se porter). The cognate Old English compound behabban meant "to contain," and alternatively the modern sense of behave might have evolved from behabban via a notion of "self-restraint." In early modern English it also could be transitive, "to govern, manage, conduct." Related: Behaved; behaving.
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.