This is (at least for me) the easiest and cheapest way to fix the home button. I had a Ipod touch 1st Gen which had a broken home button and Apple change it for free, with this second Ipod the home button also broke but I have no longer warranty so I figure out a way to fix it. Hopefully this will work for you too.
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Step 1: Cut a Piece of PVC Card
At this point you should find a regular pvc card that you dont use anymore, a bank card or and old school one may do the work. The measures that worked for me are 1.7 x 0.5 cm
Step 2: Insert It Below the Connector.
Insert the cutted piece below the ipod connector (WHERE THERE ARE NO METALIC PINS, IN THE ROW CLOSE TO THE BACK).
Rotate it and place it so it integrates with the Ipod.
Be sure to leave it long enough so you can pull it with your nail.
And thats it this trick hopefully will fix your home button as did with mine.