The origin from Etymonline indicates: 1888, probably from fag "loose piece, last remnant of cloth" (late 14c., as in fag-end "extreme end, loose piece," 1610s) This appears to be speculative, and doesn't necessarily explain why this definition fell into common usage to indicate a cigarette
for Virginia Slims Menthol cigarettes EBONY Magazine November 1991 B & H introduces cigarette specifically for women' SAN FRANCISCO, CALIF
And he very wonderfully fought for it to go in the show, and he was just so good, and he took the note that Julio had for him: You should not be indignant
You might think of these showy flowers as something of a liability to the plant, for it is they which would get picked first
Simply being around friends who are casually passing a cigarette can eventually pose serious health threats through secondhand smoke, he adds