Old English hlanc "loose and empty, meagerly slim, flaccid," from Proto-Germanic *hlanka-, forming words meaning "to bend, turn," perhaps from PIE root *kleng- "to bend, turn," with a connecting notion of "flexible" (compare German lenken "to bend, turn aside;" see flank (n.)). In Middle English, "Some examples may be long adj. with unvoicing of g" [The Middle English Compendium]. In reference to hair, "straight and flat," from 1680s. Related: Lankness (1640s).
The noun is not found in Old English, where it is represented by lank "the hip" ("turn of the body"), hlencan (plural) "armor." Meaning "a division of a sausage made in a continuous chain" is from mid-15c. Meaning "anything serving to connect one thing or part with another" is from 1540s. Sense of "means of telecommunication between two points" is from 1911. Missing link between man and apes dates to 1880.
"sereus membrane lining the chest cavity," early 15c., from medical Latin, from Greek pleuron "a rib," in plural, pleura, "the ribs, side of the body," also "flank of an army, page of a book," a word of unknown origin.