I may have found a solution. After the problem persisted through last night into today, I finally called Applecare. They ran a remote diagnostic which turned up something peculiar. Even though the phone had been updated to iOS 7.0.2. (which showed in Settings->General->About on the phone and in iTunes with the phone connected) she was seeing that it still had 7.0.1. All my software updates had to this point been OTA, and from what I understand when you do OTA updates, iOS will not replace the entire OS, but just upgrade certain compontents.
This led me to hypothesize that there was some software corruption, possibly associated with an OTA update. I backed up the phone to iTunes and then restored it in the main screen in iTunes, which causes iTunes to install a fresh verison of iOS 7.0.2. I then set up the phone as new instead of from my backup. I set up Touch ID and it worked flawlessly for several hours on all four fingerprints I registered. Of course, setting up a phone as new can cause a lot of headaches. All your texts, camera roll, call history and anything not synced to your computer, iCloud or an Exchange account, etc. are gone. You will have to reinstall all apps, set up passwords again, etc. This is a major, major undertaking and quite time consuming if you've been using your iPhone for years. There are programs out there that allow you to extract MMS/SMS/iMessage messages and photos from a backup to a computer, but they cannot be restored to the device (as far as I can tell).
Now that there is a fresh copy of iOS 7.0.2 on the device I am going to do another restore from my last backup using iTunes. I am hoping that even if I use the backup, now that there's a fresh install of the software the Touch ID will still work. If not, it seems to me the only viable option is to start from fresh and restore, set up as-new.
To me, this all but rules out a hardware issue. I am curious if anybody else having this problem had updated the OS using iTunes instead of the OTA update. Did it appear only after updating to iOS 7.0.2.? Has abybody else had this problem on a new phone with iOS 7.0.2. out-of-the-box?