Minecraft can be quite demanding, the minimum specs are a bit misleading.
If portability is not a real requirement, a desktop may be cheaper, but once you add monitor/keyboard etc that can defeat the savings somewhat. But you can also keep/upgrade the parts independently, which will be cheaper in the longer term.
At your budget, you want to be looking at second hand. New gaming laptops at staples are starting around $800. There are probably second hand bargains to be had, just look for anything with an Nvidia or AMD graphics card, and as much RAM as possible (minimum 8gb, ideally 16gb or more). Upgradable RAM might be good, but its very rare on laptops now.
I cant help on the second hand market, but perhaps its worth tasking your son with doing the research?