You can do this with something like dyndns.org. You'll also need a router that supports it (which is how I do it on my dev server), you might also be able to get software you can install in linux/windows that does this for you with dyndns.org instead of your router, though I'm not up to date on that.
Alternatively, you can update your name server address every time the IP changes. Really though, I'd recommend just breaking down and paying for hosting. You can get it very inexpensively. You'll be up nearly 100 percent of the time, it'll never drag down your home bandwidth. Also, you don't have to worry about getting popular. (If you get popular, your ISP may catch you hosting out of your own house and you could get in pretty big trouble).
I offer hosting for 2 bucks a month. That gives you mysql databases, postegre databases, python, perl, ruby, php, email, spam filters, mailman lists, and we'll soon be looking to offer svn if that's something you'd be interested in.
It's not just for BYOND you know!