I once stayed at the Heathrow Ibis. It was relatively cheap (£65 I think, though that was some years ago), but it definitely wasn't pretty. Our room was a depressing little box barely big enough to wedge the bed into. And the TV didn't work.
It wasn't a tragedy, but I didn't try the experiment again.
I have airport OCD, too. I've never slept over at Heathrow, but I've done it at Logan more than once. And Logan isn't one of those busy 24/7 airports. It's not comfy, but nobody cares if you do.