"to droop, decline in strength, become weary" (intransitive), 1520s, of uncertain origin; OED is content with the "common view" that it is an alteration of flag (v.) in its sense of "droop, go limp." Transitive sense of "to make (someone or something) fatigued, tire by labor" is first attested 1826. Related: Fagged; fagging.
British slang for "cigarette" (originally, especially, the butt of a smoked cigarette), 1888, probably from fag "loose piece, last remnant of cloth" (late 14c., as in fag-end "extreme end, loose piece," 1610s), which perhaps is related to fag (v.), which could make it a variant of flag (v.).
"put to work at certain duties, compel to work for one's benefit," 1806, from British public school slang fag (n.) "junior student who does certain duties for a senior" (1785), from fag (v.1). Related: Fagdom (1902); faggery "fatiguing labor" (1853). Brain-fag (1850) was an old term for "mental fatigue."
shortening of faggot (n.2) "male homosexual," by 1921. Fag hag "heterosexual woman who keeps company with gay men" attested by 1969.