"to droop, decline, tire," 1520s, apparently an alteration of flag (v.) in its sense of "droop." Transitive sense of "to make (someone or something) fatigued" is first attested 1826. Related: Fagged; fagging.
British slang for "cigarette" (originally, especially, the butt of a smoked cigarette), 1888, probably from fag-end "extreme end, loose piece" (1610s), from fag "loose piece" (late 15c.), which is perhaps related to fag (v.).