From this Reddit thread:

Take a screenshot or picture of your AirPods serial number. You can’t replace any parts without it.

Do this when you first get them!!

You can take a picture of the box or a screenshot from Settings > General > About > AirPods so that Apple can pull up the serial number.

If you lose the case and don’t have the serial number anywhere, you can’t even replace the case without it.

First things first, I do think it’s worth taking a screenshot of your AirPods serial number. It’s not that you can’t recover it if you lose, say, the case, but it will save you the hassle of hunting it down later.

I also think it’s interesting that the serial number only appears when you are connected to a device. So if you run out of power and lose your case, having the serial number on a screen shot will make replacing that case a bit easier.

I do believe that you can use anyone’s case to charge your AirPods. So if you lose your case, borrow one (from an Apple Store if need be) and charge up, then connect to your iPhone and look up the serial number.

All that said, here’s how to bring that serial number up on your iPhone:

• Go to Settings > General > About
• Notice that, if your AirPods are not connected, there’s no line for AirPods
• Connect your AirPods. The AirPods line should appear in the About list.
• Tap AirPods

That’s it. In addition to the Serial Number, you’ll also see the Model Number, Firmware Version, and Hardware Version. Take a screenshot, save it in Photos.