"bend of the arm," c. 1200, elbowe, from a contraction of Old English elnboga "elbow," from Proto-Germanic *elino-bugon, literally "bend of the forearm" (source also of Middle Dutch ellenboghe, Dutch elleboog, Old High German elinbogo, German Ellenboge, Old Norse ölnbogi).
First element is from PIE *elina "arm," from root *el- "elbow, forearm." Second element is from Proto-Germanic *bugon-, from PIE root *bheug- "to bend." To be out at elbows (1620s) was literally to have holes in one's coat. Phrase elbow grease "hard rubbing" is attested from 1670s, from jocular sense of "the best substance for polishing furniture." Elbow-room "room to extend one's elbows," hence, "ample room for activity," is attested from 1530s.
"thrust with the elbow," c. 1600, from elbow (n.). Figurative sense is from 1863. Related: Elbowed; elbowing.