c. 1200, from Old English hlinian "to recline, lie down, rest; bend or incline" (Mercian hleonian, Northumbrian hlionian), from Proto-Germanic *hlinen (source also of Old Saxon hlinon, Old Frisian lena, Middle Dutch lenen, Dutch leunen, Old High German hlinen, German lehnen "to lean"), from PIE root *klei- "to lean."
Transitive sense "cause to lean or rest" is from 14c. Meaning "to incline the body against something for support" is mid-13c. Figurative sense of "to trust for support" is from early 13c. Sense of "to lean toward mentally, to favor" is from late 14c. Related: Leaned; leaning. Colloquial lean on "put pressure on" (someone) is first recorded 1960.
"thin, spare, with little flesh or fat," c. 1200, from Old English hlæne "lean, thin," possibly (Skeat) from hlænan "cause to lean or bend," from Proto-Germanic *khlainijan, which would connect it to Old English hleonian (see lean (v.)). But perhaps rather, according to OED, from a PIE *qloinio- (with cognates in Lithuanian klynas "scrap, fragment," Lettish kleins "feeble"). Extended and figurative senses from early 14c. In business jargon, paired with mean (adj.) from 1970s to suggest aggressiveness as if from hunger.
c. 1200, "lean animals or persons," from lean (adj.). Meaning "lean part of anything, muscle without fat, lean meat" is mid-15c.