Check this site. They sell a replacement Home Button (currently out of stock though, but I saw a few cheap ones on ebay) and if you scroll to the bottom, they have a video on how to open it and replace it.
I would recommend that you watch the video. If you are comfortable doing this, then get the part from ebay (or any other site that sells them - just make sure it's from the 2G not 3G) and replace it yourself. If you don't feel comfortable doing this, I recommend that you let Apple handle it. There are few sites that offer mail-in iPhone service, but to be honest, I don't trust them. Stick with Apple if you don't want to do it yourself.
I hope this is helpful.