I had this problem on my iPhone 6 when I updated to iOS 10.3.
The home button did not respond upon pressing it to exit an app and return to the home screen. The device did not respond the first time, but did when pressed a second time.
If I open an app and immediately try to return to the home screen, the home button responds as expected.
If I open an app and do not actually use the app, (that is, I just leave it open) and try to go to the home screen, the home button responds as expected.
I’ve found that the only time the home button does not take me back to the home screen after a single click and requires two clicks to do so is when I’ve actually used the app I was in, (that is, I’ve navigated within the app - switching tabs, scrolling, typing and tapping things).
I tried a clean restore to iOS 10.3, using the .ipsw file, via iTunes, to no avail.
This really got to me so I DM'd @AppleSupport on Twitter. They did a hardware diagnostic to rule out any problems with my home button. They said the overall health of my phone looks good. They suggested enabling the Rest Finger to Unlock setting in the Accessibility settings. That didn't fix the problem either.
In the end, they suggested I visit an Apple Store near me. I live in South Africa - we don't have an official Apple Store yet, but we do have an 'iStore', (they're the closest thing we'll get for now). They suggested I take my device there.
Now, knowing that there isn't anything actually wrong with my home button or device for that matter, (this problem wasn't present on 10.2.1) I decided, whatever. I'm just going to restore to iOS 10.2.1 since it's still being signed, instead of wasting time going there.
Lo and behold, a clean restore to iOS 10.2.1 with the ipsw file via iTunes has 'fixed' the issue. The home button now takes me out of an app and back to the home screen with one press, as it always had, and is supposed to.
Of course, this absolutely isn't a permanent fix, as Apple's going to stop signing 10.2.1 any day now.
I spoke to someone else with this problem and apparently Apple Support had suggested to them that it may yet be a hardware problem that was just "exposed" by the update.
Honestly, since it was fine on iOS 10.2.1, maybe this can be sorted out on the software side.
I'm really hoping they take note of this and maybe try to fix it? I'm not sure.
Since then I’ve updated to iOS 10.3 yet again (because I was concerned about not having the security updates on 10.2.1) and the issue is present once again. Contemplating going back to 10.2.1 *again*, just to solve this problem until Apple fixes it. It's really annoying.