According to the Online Etymology Dictionary, it comes from "fag", meaning a loose piece of cloth:. fag (n.1) Look up fag at Dictionary.com 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 ... More @Wikipedia
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