"Boob" is a 1950's shortening of "booby", which in the 1930's came from "bubby". Linguists aren't sure, but "bubby" may derive from the German "Bübbi" which means 'teat'.
In 1347, when the bubonic plaque ravaged Europe, one of the symptoms of the plaque was called bubos or boobos which was a swelling of the lymph nodes, hence swelling of the chest can be referred to as boobs.
From the novel The History of Joseph Andrews by Henry Fielding,(1762) the Lady of the
house( Lady Booby) attempts to seduce the hero and the first thing she does is reveal her breasts, and for convoluted reasons it was also mistake not to have "taken up the offer".