The power button stopped working: it wouldn't turn on or off, and there was no click sensation when you pressed it. It felt like the button was pressed in all the time, but somehow disabled. There was still a bit of play, and nothing was corroded or dirty. It just didn't do anything when you pressed it.
The iFixit guide was invaluable for showing how to disassemble the phone and remove the power button, but it did not explain how the button and power switch actually worked. Once the actual micro switch was exposed, I realized the switch itself still worked if I pressed on it directly, it just didn't work with the button, so I ordered another button hoping that was the problem, and reassembled the phone while I waited for the new one to arrive. I verified the exposed power switch was working, then covered the opening with tape to keep the schmutz out until the replacement arrived. Unfortunately, the new button didn't fix it! There was nothing wrong with the button itself, it turned out there was a small dot on the surface of the switch that was shown in the iFixit guide that was missing. I used a minute dab of nail polish applied with a toothpick to the center of the switch, and shaped it into a tiny 1 mm dot in the center of the dome as it was drying. This worked like a charm!
The trick was realizing there was a difference between my phone's switch and the switch shown in the photo in the iFixit guide. The photo showed a tiny black dot on the surface of the yellow tape covering the tiny dome switch, and this was missing on my phone. This dot makes contact with the center of the small cylinder in the back of the button, and sets the distance the button travels when it's pressed. Without it, the button just bottoms-out and doesn't actually push on the dome switch. I wish the ifixit guide explained this.