This is an interesting option and I really appreciate your contribution. Deleting a branch is a somewhat destructive operation (it can be restored but not obviously with the click of a button) so I'm hesitant to include this in the stale bot.

@hross Happy for you to close this if you decide against including it. My understanding was you can restore a branch using the GitHub UI.

Not sure if the button is a permanent feature, or if it eventually expires, but my understanding of GitHub is that a PR diff with a deleted branch is available indefinitely?

Edit:

I just tested this in the example PR by restoring the branch using the "Restore branch" button and reopening the PR alexbrazier#1

I would also like this feature. We make a lot of PRs and many of which go stale so this would help us keep the number of useless branches down and our repo more efficient.
As it's an option, people can choose to turn it on and accept the risk :)