fag end n.
1. [17C+] (also butt-end) the last part or remnant of anything.
2. [19C+] a fragmentary part of a speech or conversation that one might overhear, just as it tails off.
3. [mid-19C+] (also fag) the butt of a cigarette or cigar.
[1910s–20s] a man who collects cigarette ends from the pavement.
[1910s+] to listen in to other people’s conversations and attempt to comment upon them or join in; esp. as don’t pick up fag-ends.