So I'm looking at moving my VPS into a box at home, but I have an IP that changes every so often. What's the best way to fix that? I've got no problems with DNS being hosted on Route53 or something else.
Yep. They give you a little program that runs on your computer that checks your external IP address at frequent intervals and updates the DNS records when it changes. Better yet, some home routers have a configuration page where you can select among popular Dynamic DNS providers and then you don't need to run the proprietary program. My ISP-provided router (FiOS) has ZoneEdit in the list which I've used for years even before I had FiOS, so it was a nice surprise when I found it there. Just for sake of example here's a link: http://www.zoneedit.com/dynamicDNS.html
Some registrars even provide such a service for free (well, included in the annual registration fee), though look for reviews to see how reliable your registrar's is before trying it in anger.