if I could order it off Amazon, that would be great, but I'm open to any suggestions.

I purchased a Beelink SER5 Mini PC with AMD Ryzen 5 for my mother. It has 6-cores/12-threads. https://www.amazon.com/dp/B0BCQKFP47.

You can get the SER5 for about $310. It is a reasonably priced machine. I got it little cheaper because I bought it during the Thanksgiving sale last year.

When I benchmarked it, it was faster than my Intel Core-i5 10th gen. The machine is way more than my mother needs. I was kind of envious because I paid about $200 more for the Intel machine.

It would be mostly for school work, but also some games: they really like Minecraft and retro games

The GPU has 6 cores. It should perform moderately well for [well written] games. But I am not a gamer, so I never tested performance.

Maybe a mini PC might be good, so I could kind of hide the box behind the monitor or something.

Yeah, I think I saw mounting hardware to hang a cube behind the monitor. But I have never used it. I think it also depends on the monitor.