I'm absolutely certain that ScottNY71, dubzau and a couple of others are right. This is a hardware issue that iOS 8 doesn't play nicely with, but iOS 7 was more forgiving about.
I had this problem with an iPhone 5, and replacing the home button ribbon cable fixed it completely. I have had no problems since, not a single accidental double-click.
I'm sure it's no coincidence that this is older, well-used phones that have the problem. The home buttons are wearing out. All of the "fixes" that have been posted pretty much involve using the home button a bit -- which also made mine work more reliably for a while! It probably just temporarily dislodged the dust or whatever was causing the problem.
i guess the good news is iOS 8.1 is reported to be more forgiving again, but I'm pretty sure anyone who had this problem can expect further signs that the button is wearing out in future. Not that I'd necessarily recommend doing the ribbon replacement surgery yourself, it's tricky! But it does work.