You'll probably have to get ahold of AppleCare, or visit an Apple Store to see what they can do to fix the problem you are having. However, you can do the following as a workaround until it's fixed...
Go to Settings -> General -> Accessibility -> Assistive Touch -> and turn that ON. You'll now have a small white dot on your screen that you can use for all the features that you would use the Home button for. You can also move it around to different places, and it dims when it's not in use.
My wife had this problem on her iPhone 4 and it allowed her to continue using her iPhone until it was fixed.
MLadd, your suggestion worked perfectly. I have an iPad 1, first generation, and I first had to plug it into my Apple Computer and update the operating system. That was easy to do. The moment my iPad was plugged in, the iTunes menu came up and gave me the option of updating the operating system which I did to 5.1.1 Now I was able to access the Accessibility and Assistive Touch menus, which the old original operating system didn't allow me to do.
By the way, we both put the dot on the screens of our iPhones as well, to reduce the wear and tear on the home buttons on both phones.