I would have thought 'fag end' was a relatively modern word. However, I've just seen it in my wonderful book from 1688, "Epistolae Ho Elianae Familiar Letters Domestic and Forren" (previous post)
A quick search reveals:
|n.||1.||An end of poorer quality, or in a spoiled condition, as the coarser end of a web of cloth, the untwisted end of a rope, etc.|
|2.||The refuse or meaner part of anything.
The fag-end of business.
And it turns out to have been in use for around 400 years:
And fag is a corruption of flag:
(If that's too small to read, here's the link)