How can I make my iPod Touch home button more responsive?
The unresponsive iPhone/iPod Touch home button is a known problem. I had an iPhone which started with the same problem as yours. At first the home button worked with just a soft touch, and from any angle. Then it started to work only with a hard press and not at all from certain angles. Finally, it either completely ignored presses, or responded to a single press as if it had been pressed twice (multitasking bar appeared).

It was still inside the one-year warranty, but I bought AppleCare anyway, figuring that it might recur. I took it to an Apple Store and the guy played with it for a minute, told me I had a faulty device and gave me a brand new iPhone. The home button can't be replaced, so they just give you a new one.

My advice it to take it to an Apple Store if there's one nearby, or if not call Apple and report the problem. Most likely you'll get a new, working iPod Touch.