I just want to report that I have the same exact problem as the OP. Brand new iPad4, 64gb wifi only. The problem is present right out of the box, so this isn't due to wear and tear. I've tried all of the suggested solutions. They are only temporary. But surely the fact that there *can* be a temporary solution with these recalibration tricks, even if it only lasts 30 seconds (my average), means it's a software problem?
With gestures available, I wouldn't care about the home button at all... except for the all-important need to get icons out of jiggly-mode once you've repositioned them. Only the home button can effect that transition, and it's such a frequent need that this problem really *is* frustrating, especially for such an expensive device.
Do you guys really recommend that I make the trek to the Apple Store? Should I go through the hassle of a replacement if that's going to exhibit this problem as well? Aren't these places swamped right after Christmas? I think my two-week, 'no questions asked' period is up by next Wednesday.