Paul Erdős was famous for offering money for solutions to math problems. My understanding is that those prizes are still being administered by Ron Graham, even though Erdős passed away several years ago.

One of his most famous (and biggest money) problems is the following:

Conjecture: If $S$ is a set of positive integers such that the series $\sum_{s \in S} \frac{1}{s}$ diverges, then $S$ contains arbitrarily large arithmetic progressions.

I believe that \$3000 is offered for a proof.

Several graph theory related problems, with prize money listed for many of them, are collected in Erdős on graphs, by Chung and Graham.