c. 1300, "harm, injury; hurt or loss to person, character, or estate," from Old French damage, domage "loss caused by injury" (12c., Modern French dommage), from dam "damage," from Latin damnum "loss, hurt, damage" (see damn). In law (as damages) "the value in money of what was lost or withheld, that which is given to repair a cost," from c. 1400. Colloquial sense of "cost, expense" is by 1755. Damage control "action taken to limit the effect of an accident or error" is attested by 1933 in U.S. Navy jargon.
"soft, grayish mass filling the cranial cavity of a vertebrate," in the broadest sense, "organ of consciousness and the mind," Old English brægen "brain," from Proto-Germanic *bragnan (source also of Middle Low German bregen, Old Frisian and Dutch brein), a word of uncertain origin, perhaps from PIE root *mregh-m(n)o- "skull, brain" (source also of 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. The figurative sense of "intellectual power" is from late 14c.; the meaning "a clever person" is recorded by 1914.
To have something on the brain "be extremely eager for or interested in" is from 1862. Brain-fart "sudden loss of memory or train of thought; sudden inability to think logically" is by 1991 (brain-squirt is from 1650s as "feeble or abortive attempt at reasoning"). 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."