"stupid person," 1590s, from Spanish bobo "stupid person," also used of various ungainly seabirds, probably from Latin balbus "stammering," from an imitative root (see barbarian).
Specific sense "dunce in a school class" is by 1825. Booby prize "object of little value given to the loser of a game," is by 1884:
At the end of every session the dominie had the satirical custom of presenting his tawse [a corporal punishment implement used for educational discipline] as a "booby-prize" to some idle or stupid lout whom he picked out as meriting this distinction so that next time they met he might start fresh and fair with new pair for a new set of classes. [Ascott R. Hope, "Dumps," Young England magazine, Sept. 1884]
Booby trap is by 1850, originally a schoolboy prank; the more lethal sense developed during World War I. Booby-hatch "wooden framework used to cover the after-hatch on merchant vessels" is from 1840; as "insane asylum" by 1936.
"breasts," 1929, U.S. slang, probably from much older term boobies (late 17c.), related to 17c. bubby; perhaps ultimately from Latin pupa, literally "little girl," hence, in child-talk, "breast." Or else it is a natural formation in English (compare French poupe "teat," German dialectal Bubbi, etc.).