How can I set up a from home hosted website without buying a static IP?

So I know that somesites allow you to host web content for you, but they can start to charge an arm and a leg based on how much you want to update it, how much you have, and other things. Pretty much what I want to do is host a personal site for and about me. Essentially access things I need like RPG characters and references, cosplay stuff, gaming stuff, and all that ilk.

I'm going to be investing in a small box that is more or less always on and always online, unless I loose internet for some reason or have to restart a router etc, but I'm running into the issue of I need to buy a static IP which is around $60 a month. That's $60 I don't have or want to spend on a little pet project. And before you suggest or ask, no I don't want just a site like blogger or w/e to do it for me.

My question is, and I know that I've been told this before by a friend I just can't seem to remember what he said, is there a way this can be done without using a static IP address and hosting the site from home still?

Oh I should also add that I'd eventually like this to be something I can host my own RESTful webservice on for small little one off personal apps I'd use for myself too.