Upgraded to 6.1 and had the same problem.
After a long frustrating time I started wondering if it might be something related to accessibility... So I went into system settings, general, accessibility (my first time exploring this area of the settings actually), home-click-speed (which seemed most promising).
Initially I set it to slow just to see if it works... the home button was now behaving reliably, but of course, it behaved slowly (since it has to wait longer to determine how many clicks you are going to do). For no apparent reason, I then switched to slowest (though I'm sure you don't actually have to do that, just mentioning it for completeness). After confirming that my home button was now working correctly, I set it back to default.
End result: It now works perfectly, very responsive and no different to how it was before upgrading to 6.1. I even restarted the iPad to confirm it wasn't just a fluke. As far as I can tell, it's fixed.
I decided to post this since the other home button fix solutions online just didn't seem to work. Anyway, hope this will help out at least a few people who are running into this problem.
- Raz