Old English hit, neuter nominative and accusative of third person singular pronoun, from Proto-Germanic demonstrative base *khi- (source also of Old Frisian hit, Dutch het, Gothic hita "it"), from PIE *ko- "this" (see he). Used in place of any neuter noun, hence, as gender faded in Middle English, it took on the meaning "thing or animal spoken about before."
The h- was lost due to being in an unemphasized position, as in modern speech the h- in "give it to him," "ask her," is heard only "in the careful speech of the partially educated" [Weekley]. It "the sex act" is from 1610s; meaning "sex appeal (especially in a woman)" first attested 1904 in works of Rudyard Kipling, popularized 1927 as title of a book by Elinor Glyn, and by application of It Girl to silent-film star Clara Bow (1905-1965). In children's games, the meaning "the one who must tag or catch the others" is attested from 1842.
From Old English as nominative of an impersonal verb or statement when the thing for which it stands is implied (it rains, it pleases me). After an intransitive verb, used transitively for the action denoted, from 1540s (originally in fight it out). That's it "there is no more" is from 1966; this is it "the anticipated or dreaded moment has arrived" is from 1942.
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]
by c. 1200 as an emphatic form of Old English of (see of), employed in the adverbial use of that word. The prepositional meaning "away from" and the adjectival sense of "farther" were not firmly fixed in this variant until 17c., but once they were they left the original of with the transferred and weakened senses of the word. Meaning "not working" is from 1861.
Off the cuff "extemporaneously, without preparation" (1938) is from the notion of speaking from notes written in haste on one's shirt cuffs. In reference to clothing, off the rack (adj.) "not tailored, not made to individual requirements, ready-made" is by 1963, on the notion of buying it from the rack of a clothing store; off the record "not to be publicly disclosed" is from 1933; off the wall "crazy" is 1968, probably from the notion of a lunatic "bouncing off the walls" or else in reference to carom shots in squash, handball, etc.