IMO your best bet at that price range would be laptops with an AMD APU as the integrated GPU is really good for moderate gaming. If you live in the US, there's for example this auction on Ebay (currently at $241.50): https://www.ebay.com/itm/355230104508 . For that particular laptop, I don't know about the drive slots but apparently it at least has one slot of upgradeable RAM so that's a big positive I guess: https://www.crucial.com/compatible-upgrade-for/asus/vivobook-16x-m1603
You can start from there and look for similar specs on Ebay. When in doubt, just copy the name of the integrated GPU and search "[whatever your son is playing] + [the name of the GPU]" on YouTube, or just look for generic benchmarks.