It's a term value investors use for investing in "net nets", i.e. companies that trade below their net cash position. The idea is "if we can buy the cash at a discount and get the (crappy) company for free, at some point the market will pay at least for the full cash value".

The approach was made famous by Benjamin Graham and used by Warren Buffet in the first phase of his investing life.

Nowadays it's harder to find "net nets" as everybody and his dog can run a screen to find those companies now, so they get bid up above their net cash position most of the time.

Most people throw away the cigar butt, I imagine cigar butt investing is picking up other people's leftovers.