Should I share the prize money from a paper award for my graduate work with my former PhD advisor?
I suggest splitting the prize money 50-50 with your advisor, or at least offering (with honest intent) to do so. It is the magnanimous and appropriate thing to do, especially considering that his mentorship and assistance to you in turning the project into a finished, polished paper are likely to have been instrumental in getting the award. And there is always the chance that he will be even more magnanimous and refuse to claim his share... By contrast, trying to come up with a formula for quantifying each author's contributions seems petty and ultimately futile. After all, the work may have been "entirely yours", but the paper carries both of your names, and from your description it sounds like the award is for the paper. Logically, I'd say the adviser definitely deserves at least a share, and very probably deserves an equal share.

Finally, even from a purely utilitarian (some would say cynical) point of view, by making such an offer you are likely to impress your adviser (and others who will hear about this -- word gets around!) as a sensible and pleasant-to-work-with person, which would be good for your reputation and could end up benefiting your career trajectory in all sorts of unexpected ways.

Congratulations on the award!