c. 1300, "knob or ball attached to another body," especially as used to hold together different parts of a garment by being passed through a slit or loop (surname Botouner "button-maker" attested from mid-13c.), from Old French boton "a button," originally "a bud" (12c., Modern French bouton), from bouter, boter "to thrust, strike, push," common Romanic (cognate with Spanish boton, Italian bottone), ultimately from a Germanic source, from Proto-Germanic *buttan, from PIE root *bhau- "to strike." Thus a button is, etymologically, something that pushes up, or thrusts out.
Meaning "point of the chin" is pugilistic slang, by 1921. A button as a round protuberance you depress to create an effect by closing an (electrical) circuit is attested from 1840s. Button-pusher as "deliberately annoying or provocative person" is attested by 1990 (in reference to Bill Gates, in "InfoWorld" magazine, Nov. 19). In the 1980s it meant "photographer."
The meaning progressed from "compliant, obliging," through "lively, jolly," "healthily plump, vigorous and attractive," to (in women, and perhaps influenced by lusty) "attractively plump, comely" (1580s). In Johnson [1755] the primary meaning still is "obedient, obsequious." It was used especially of women's figures from at least 1870s, and by 1950s it had begun to be used more narrowly for "bosomy" and could be paired with slim (adj.). Among its cognates are Dutch buigzaam, German biegsam "flexible, pliable," which hew closer to the original English sense.
Old English bycgan (past tense bohte) "get by paying for, acquire the possession of in exchange for something of like value; redeem, ransom; procure; get done," from Proto-Germanic *bugjan (source also of Old Saxon buggjan, Old Norse byggja, Gothic bugjan), which is of unknown origin and not found outside Germanic.
The surviving spelling is southwest England dialect; the word was generally pronounced in Old English and Middle English with a -dg- sound as "budge," or "bidge." Meaning "believe, accept as true" is attested by 1926. Related: Bought; buying. To buy time "prevent further deterioration but make no improvement" is attested from 1946.