have on the brain

Old English brægen "brain," from Proto-Germanic *bragnam (cf. Middle Low German bregen, Old Frisian and Dutch brein), from PIE root *mregh-m(n)o- "skull, brain" (cf. Greek brekhmos "front part of the skull, top of the head"). But Liberman writes that brain "has no established cognates outside West Germanic ..." and is not connected to the Greek word. More probably, he writes, its etymon is PIE *bhragno "something broken."

The custom of using the plural to refer to the substance (literal or figurative), as opposed to the organ, dates from 16c. Figurative sense of "intellectual power" is from late 14c.; meaning "a clever person" is first recorded 1914. Brain teaser is from 1923. Brain stem first recorded 1879, from German. Brain drain is attested from 1963. An Old English word for "head" was brægnloca, which might be translated as "brain locker." In Middle English, brainsick (Old English brægenseoc) meant "mad, addled."