If you are doing this for the experience or fun of it, rock on.
If you are doing this because you want to have a reliable website, don't bother. Homes make lousy datacenters for the same reason that website hosting costs money: ensuring the power stays up, the net stays up, the server and the applications are up is capital and labor intensive which is most efficiently amortized across a server farm.
Some routers support a "roaming" DNS server like http://www.dyndns.com/ which allows you to maintain a static name even though your ISP may shuttle your IP address around dynamically, but the caveat of "this may irritate your ISP" does apply.
(disclosure: I have a free DynDNS account for no really good reason, it works on the odd times I've tried it, I don't use it for anything important, my ISP is very liberal and I have no connection to DynDNS aside from that.)