You can try the following that helped me fix my homebutton thing on my iPhone 4. But WARNING - it only fixes MECHANICAL issues, not software issues! If this is not your case or your button is really physically broken, don't try this at home and downvote me! :-)
- Switch off the phone and pull out your SIM card.
- Buy medical alcohol in a pharmacy with a very high percentage, at least 97% (in Germany it is called Isopropanol or Isopropyl alcohol).
- Take a pipette and drop a few drops onto the home button, so the button is completely covered in it.
- Press and release the button at least 50 times, starting slow and increasing speed.
- Add more alcohol, but less and carefully, only a few drops! don't drain your phone!
- Start pressing and releasing again at least another 50 times.
- It should be very easy now to press the button.
- Let the iPhone dry and recover for at least 8h.
- Put in again your SIM card and power on the phone.
- Feel happy with a recovered home button. :-)
After your home button is working again, try again your previous steps for recovering your phone.