Do you have any parts such as a case, PSU, monitor etc. that you already own or do you have to buy everything?
What kind of performance do you/he expects and which games does he want to play?
I don't know what you can buy for 500 dollars that would actually run somewhat recent games at reasonable performance but you can get close.
CPU: AMD ryzen 2600 (150)
Main board: ASrock b450 (60)
Memory: 2x8gb Corsair vengeance 3200mhz cl16 (65)
GPU: gtx 1660 (210)
Storage: 512gb SSD (65)
That is 560 dollars. Add another 100 if you need a case and PSU and another 100 ~ 150 if you also need a monitor plus another 50 if you also need a mouse and keyboard.
That system will run games but don't expect any high settings or fps. To be honest I think right now is a bad time for budget gaming PCs.
If you want to game onna budget you are much better off buying a ps5 or Xbox series x. The value proposition is just much better.
If it's more about building something with your kid you might get something old and second hand for 50 bucks or whatever and do some kind of retro machine?