Kenneth G. Wilson (1923). The Columbia Guide to Standard American English. 1993.
fag end
is a cliché, a frozen figure whose origins are almost forgotten: the fag end of a piece of cloth or rope is frayed, worn, and worth little. Hence the fag end of a difficult day. It is thought to come from a Middle English word meaning flap, but it may in some way be related to the verb meaning to exhaust, to wear out. In British slang, a fag end is a cigarette butt.