You can do what you want with a ‘dynamic DNS’ service. That gives you a host name which can refer to your home’s dynamic IP address. You run a piece of software on your PC or router that signs in to the dynamic DNS service and keeps it updated with your current IP address. Totally doable.
You may have a great router, especially if flashed with DD-WRT. If it is, then you can set it up to automatically update your dynamic DNS account, and not rely on your PC to do it. That’s how mine is set up.
Here’s the dynamic DNS service I use (I use a free account).
And, here’s the link to DD-WRT (if you are interested, as it is optional). You’ll have to see if your particular model of WRT54G is supported or not.