The power button on my iPhone 4s stopped working, all of a sudden and for no apparent reason. I set it on my desk after work one day and when I came back about an hour later the button wouldn't click. Fortunately it would still make electrical contact but only with a lot of pressure.
Fixed! Not exactly an easy task but do-able thanks to the guides on iFixIt, and far less costly than going to Apple or a 3rd party repair shop.
On the power button's electrical switch there is a small plastic dot about the size of a . that had become dislodged. The manufacturing tolerances are so fine that the difference of <0.5mm between the metal power button and the electrical click-switch was enough that it wouldn't click.
So I repositioned the small bit of plastic and used a minuscule amount of superglue to hold it in place. After screwing in the power switch, it worked normally once again.
I did get a rather nasty surprise after reassembling the phone: the display didn't work, ack! The phone was working properly though Siri didn't have any advice for "why doesn't the display work?". So I disassembled the phone again, re-seated the display cable and fortunately that fixed the problem.
The best advice is already in iFixIt's guide: if you're not comfortable, experienced, and skilled in small electronics repair then don't take apart your iPhone.
Two things I would add to the guide:
--Use a magnifying glass of some sort, iPhone parts are tiny. I used a 10x hand lens while supergluing the plastic dot back to the switch.
--Use a piece of tape, upside-down on your desk, to hold and organize the screws. Or at least have some method that works for you.