How old is the phone? Is it still under manufacturer's warranty? You can take it to a Verizon store or an Apple store and see what they recommend.
** I also found this advice on the iPhone boards:
Try re-calibrating the button:
Step 1: Navigate to, and launch one of the stock applications on your iPhone. If you don’t already know, thestock apps are things like weather, YouTube, stocks, settings etc (any that come as part of iOS).
Step 2: When the app is launched, press and hold the Power button located at the top of the device until the ‘Slide to power off’ appears.
Step 3: Press and hold the iPhone home button. Keep the button depressed until the Slide to power off slider disappears and the application quits back to Springboard.
Step 4: Voila! Your home button is now re-calibrated and should be more responsive for you.