One thing you may need is a dynamic DNS provider.
After registering one of these services, would be necessary to install an updater software to modify the DNS record automatically when the dynamic IP changes. If you have a router, check on the configuration page if there are a DDNS section to set up your dynamic DNS configuration and make the update automatically from there.
There are several free dynamic DNS providers and updater clients for any OS, only lack free support for CNAME option (that is fully available through commercial packages, this is required if you want to use virtual hosts, a.k.a. site1.mydomain.ath.cx).
So, this is a common issue when you want host a website or setup a development/testing server from home.