There is no fix. It's not a software issue. If it were a software issue, it would affect all iPad 4's being produced, and all iPad 3's and iPad 2's upgraded to iOS 6. But it doesn't. It only affects a (significant) fraction of them. If it affected all of them, this problem would be as pervasively reported as the Apple Maps app fiasco, which affected all. You'd be reading about it at Ars Technica, Wired, NY Times, etc.
The lameness of the Apple ("we'll replace Google") app for Maps affected everyone, which is why the above media outlets (among many others) covered it. That was a software issue. The home button issue is not like this. It only affects some but not all iPads which have iOS 6.
In my view, Apple can only fix this by better quality control at the factories.
My initial iPad 4 had buttonitis right out of the box. My replacement iPad 4 did not. There was no difference between them in terms of software - they were both straight from factory. So it was a hardware problem. And no software trick that I know of gives a lasting solution to the problem.
I've offered some theories about how the underlying hardware issue might be triggered by software, here and here. But I admit those are just speculative theories, and I'm clearly not an expert in this area. But it seems pretty clear to me that hardware differences of some sort are the relevant factor here. Apple just isn't doing a good job manufacturing many of these home buttons.
The best option for those under warranty is !Braggy!'s step 5 above - reproduce it in the Apple Store, and hopefully they'll replace it.