fag (v.) Look up fag at Dictionary.com
"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.
fag (n.) Look up fag at Dictionary.com
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.).