A bit of my mind, as “Iʹll tell him a bit of my mind,” Iʹll reprove him. Same word as bite, meaning a piece bittenoff, hence a piece generally. (Anglo-Saxon, bitan, to bite.)
Not a bit, or Not the least bit. Notatall; not the least likely. This may be not a morsel, or not a doit, rap, or sou. “Bit” used to be a small Jamaica coin. Westilltalk of a threepenny-bit. Bit, of course, is the substantive of bite, as morsel (French morçeau) of mordre.