I think you will need a dynamic dns solution. Try something like DynDNS or FreeDNS.
FreeDNS is more free. What I mean is, you get more features with the free product than with DynDNS. I've used both services, and they both work fine.
If you're not familiar with these types of services, this is how they work:
- First, set up the DNS settings
- Second, set up a script to update DNS settings when the IP changes
- Third, configure subdomains
I think you can piggy-back on someone else's domain if you like. I bought my own and got it configured in an hour or so using this guide.