I made an appointment with the Apple folks for this morning and found that Apple issued a recall on some iPhones yesterday. Talk about timing! They’ll fix the problem for free, but you’ll be without a phone for a few days unless you get a loaner. Fortunately I have Apple Care on my phone and they replaced it with a new one this morning.
When I got home from the mall, I found this article about the recall on pcmag.com: Apple to Fix iPhone 5 Sleep/Wake Button Issues for Free
If you’re having a problem with the button, you can get it fixed.