Can I find my AppleID on a broken phone?

Apple ID's are almost always a users email address. Does he have an email address?

See the following two support articles for help:

If you forgot your Apple ID - Apple Support

If you forgot your Apple ID password - Apple Support

Typically an Apple ID is an email address you have used. If you are currently using a mobile device that is logged into your account, go to Settings and tap your name at the top.

Look in Keychain, and in a browser's security settings if you use it to automatically sign into Apple when using a browser.

Try the methods mentioned earlier.

If you still need help:

Contact Apple for help with Apple ID account security "This article provides country-specific Apple Support contact information for customers seeking help with their Apple ID password or other security-related issues."

Contact Apple for support and service - Apple Support

For help with For help with an invalid account you need to contact Apple.

To reach Apple Support for an account issue:

1. Click here --> https://getsupport.apple.com/?caller=cups&PGF=PGF63005

2. Select: Disabled Apple ID

3. Select: ”The Apple ID is not active" alert

4. Select: “Continue”

5. Select a chat or telephone call option.

or:

Contact Apple for support and service by telephone --> Contact Apple for support and service - Apple Support

The only way Apple will help you is if you can present the original purchase receipt from Apple, which clearly shows the iPhone Serial Number on the receipt. Can you provide that?

And when your son get's a new phone, use Parental Controls on it and lock things like his ability to change the Apple ID on his phone, to avoid ever having to go through this again.

See how here --> Use parental controls on your child's iPhone and iPad - Apple Support

Your son will lose ALL his data he had on the now broken phone. All apps are forever tied to the Apple ID they were downloaded under. He'll have to start all over on the new phone and his photos, notes, files etc., will be gone.