The home button is installed in the mid frame assembly that holds the LCD and Digitizer. The home button has two small pins that sit on top of the Dock connector when the screen is installed which is how the home button makes contact and works.
If the pins are bent or broken....you would need to pop the lcd out, heat and pry the glass off the frame and then replace the home button. If the pins are in good order....you can test to see if its a Dock issue but opening the screen up, but leaving it plugged in. While the phone is on...use a small screwdriver to touch one of the metal pads on top of the dock where the home button pins make contact....and touch the shaft of the screwdriver to the metal (chrome color) rim around the back housing. If it makes the home button actuate on the phone...then the issue is with the button itself and you should just replace the button. If it wont even actuate the home button activity on the phone from the dock....then the dock is bad and needs replaced. You can try cleaning those contacts and pins first, but normally it just needs replaced.
Hope this helps