Old English lama "crippled, lame; paralytic, weak," from Proto-Germanic *lama- "weak-limbed" (source also of Old Norse lami "lame, maimed," Dutch and Old Frisian lam, German lahm "lame"), literally "broken," from PIE root *lem- "to break; broken," with derivatives meaning "crippled" (source also of Old Church Slavonic lomiti "to break," Lithuanian luomas "lame").
In Middle English especially "crippled in the feet," but also "crippled in the hands; disabled by disease; maimed." Figurative sense of "imperfect" is from late 14c. Sense of "socially awkward" is attested from 1942. Noun meaning "crippled persons collectively" is in late Old English. To come by the lame post (17c.-18c.) was an old colloquialism in reference to tardy mails or news out-of-date.
"to make lame," c. 1300, from Proto-Germanic *lamejanan (source also of Old Saxon lemon, Old Frisian lema, Dutch verlammen, German lähmen, Old Norse lemja "thrash, flog, beat; to lame, disable"), from the root of lame (adj.). Related: Lamed; laming.
1761, "any disabled person or thing;" especially Stock Exchange slang for "defaulter."
A lame duck is a man who cannot pay his differences, and is said to waddle off. [Thomas Love Peacock, "Gryll Grange," 1861]
Sometimes also in naval use for "an old, slow ship." Modern sense of "public official serving out term after an election" is recorded by 1863, American English. The quote attributed to President Lincoln ("[A] senator or representative out of business is a sort of lame duck. He has to be provided for") is from an anecdote of 1878.
It is well known to everybody who knows anything of its history, that this court [Court of Claims] was made a sort of retreat for lame duck politicians that got wounded and had to retreat before the face of popular condemnation. That is just exactly what it was for, a safe retreat for lame ducks; and it was so filled up; (etc.) [Sen. John P. Hale, New Hampshire, Congressional Globe, Jan. 12, 1863, p.271]