If the standard troubleshooting steps do not resolve this including restoring your iPhone as a new iPhone or not from your iPhone's backup, more than likely your iPhone has a hardware problem.
If you want another original or first generation iPhone, Apple offers an out of warranty or not covered under the warranty "repair" price, which pays for an exchange. The cost is $199 plus tax and does not require signing a new contract with AT&T.
http://www.apple.com/support/iphone/service/faq/#warranty7
If you aren't interested in doing this, you can check with a 3rd party repair service, but depending on which component or components needs to be replaced, it may not be much less than $199 which pays for an exchange.
If not, live with it or purchase a different phone.