"piece of lace worn around the neck," 1680s, agent noun from tuck (v.). In Middle English tukere was "one who dresses or finishes cloth," hence the surname.
"to tire, weary," 1833, New England slang, of uncertain origin, perhaps from tucked (past participle of tuck (v.)), which had, in reference to dogs, a slang sense of "exhausted, underfed." Especially with out. Related: Tuckered; tuckering.