The kids had used old iPhones as IPod touches without trouble, but apparently I'd never done a factory reset.
If you do that, the phone can't be used -- until it's activated.
You need a SIM card that was either originally used to activate the phone, or that works with a currently active phone:
Follow these steps to use your iPhone without a wireless service plan:
Insert the SIM card from your new, activated iPhone or one that was previously used to activate the original iPhone.
Connect the iPhone to iTunes on a computer connected to the Internet.Once iTunes activates the device, you're free to use the iPhone as if it were an iPod touch.
... It may be necessary to repeat this process after updating or restoring the device.
Yikes. If I'd known that then, when I got my 4S, I'd have asked to keep the old SIM card and had them give me a new SIM. Instead we just swapped the old SIM. (Not sure if AT&T will do this, it's an advantage of ordering you iPhone from Apple for home delivery.)
Lesson: When you upgrade your iPhone, keep the SIM card that was previously used to activate the device.
Update: On the brighter side, iOS 5 allows use of FaceTime and iMessage without a SIM card. You can't use them in airplane mode however.
Update 11/30/2011: iMessage behavior is flaky in a SIM-Less iPhone. More on that later. In a separate experiment on another phone I found a SIM card that had previously been used in a different iPhone worked to get pasts the iTunes check even though the number no longer existed and the SIM card was no longer valid.
Update 3/5/2012: iMessage was much better by mid-December and has worked well since.