Bountysource.com offers this exact service and integrates heavily with the Github API. You can load up any open GitHub issue and put a bounty on it. The bounty is then paid out to the developer whose PR is merged and closes the issue.
We also have a plugin that committers can install that automatically updates issues with bounty details when they exist (that we don't think clutters the Github UI). Example here: https://github.com/jshint/jshint/issues/28.