You won't find the Windows product key on the install disk. It doesn't work like that. Think about it. If Microsoft included the key on the disk they'd have to make each disk (of millions) unique. Also it would be like printing the PIN on a bank card. Anyone who got hold of the disk could find the key and install the OS.
To install, what is needed is not "the" unique product key for that CD, (there isn't one), but rather "a" product key that would work with that disk. Microsoft make the disks so that they will work with a number out of a range, so that there is a range of valid XP Home numbers, another range of valid XP Pro numbers, etc. The numbers go on the stickers that get stuck on the packaging, and sometimes on or inside the computer case if the machine is bought prebuilt from a retailer.
As Geezedave says, once the OS is installed, the product key that was typed in by whoever installed the OS is stored on the system. Prior to XP, this was in the registry and could be easily seen, but XP and later you need a tool such as Magical Jelly Bean Key Finder, which you can find by using Google. If Windows XP came packaged with the computer and was not the retail version (with box), the key recovered by such a tool may not be your own machines key but the OEM's bulk install key. If you have manually entered a product key this will be the one returned.
You should be aware of certain issues. If your son's computer still has XP installed from that disk, it would not be "legal" for you to make a duplicate install on another computer. It is not the policy of Computerhope to give advice on breaking Microsoft's End User Licence Agreement, other than to say "Don't do it!" So that's one issue.
Another is if you use a key which although valid technically (i.e. it works) for that OS install, the machine might fail a Windows Genuine Advantage check when you tried to apply to download a Service Pack or security updates from Microsoft. You can read about WGA on Wikipedia. Again, such an issue would be outside the scope of this forum. Disabling WGA is also something we don't advise about, I think. Service Packs and security updates are really important so this is something to think carefully about.