c.1300 (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," common Romanic (cf. Spanish boton, Italian bottone), ultimately from Germanic (see butt (v.)). 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 something you push 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."