It's probably either the home button itself, the connection to the dock connector assembly or a logic board issue. Snapping one of the prongs off on the wi-fi antenna won't affect the home button.
First, try to clear out the dock connector on the bottom of the phone. Sometimes dust/lint can get in there and cause issues. Using compressed air works great.
If you have a multimeter, you can remove the home button assembly and test the button to see if it is working properly. Just measure the resistance (Ohms) when the button is depressed. That will tell you if the home button is working by itself.
My guess would be that it is either the home button's connection to the dock connector assembly or the home button itself is broken.