limp (v.) Look up limp at
"move with a halting or jerky step," 1560s, of unknown origin, not found in Old or Middle English; perhaps related to Middle English lympen "to fall short" (c. 1400), which probably is from Old English lemphealt "halting, lame, limping," the first element of which is itself obscure.

OED notes that German lampen "to hang limp" (Middle High German limphin) "has been compared." Perhaps it is from a PIE root meaning "slack, loose, to hang down" (source also of Sanskrit lambate "hangs down," Middle High German lampen "to hang down"). Related: Limped; limping. Limpen in Middle English was a different verb, "to happen, befall, fall to the lot of," from Old English limpan, which might ultimately be from the same root.
limp (adj.) Look up limp at
"flaccid, drooping, lacking stiffness or firmness," 1706, of obscure origin, apparently from the first element in Old English lemphealt (see limp (v.)). Related: Limply; limpness. A limp wrist as indicative of male effeminate homosexuality is from 1960.
limp (n.) Look up limp at
1818, from limp (v.).