A lot depends on what you/he aim for. If you want the best possible gaming experience then you may choose to spend a high proportion of the budget on GPU and CPU. On the other hand, you might think that it's worth going for a decent CPU with good graphics on board. Things like the Ryzen 7 5700G or i7 11700K (or better) may be fine and give plenty of saving time until Christmas and graphics card season.
Worth keeping an eye out for companies (like Amazon) that have open-box offers, usually stuff that's good as new but returned because of buyer's remorse, or whatever. I got my last mobo for half price (although that saving is rare, and also I did discover that the original purchaser had obviously tried to force the USB 3 cable from his case in backwards as the pins were mangled. Probably explains the return, but I managed to straighten them).
I like your "jigsaw puzzle" idea, particularly as you'll usually get a far better deal on a used full computer than on buying used parts individually. A quick check on ebay showed a dealer selling PCs open-box but never used between £400 and £450 including a 27 inch screen. That said, I do agree with what the others say about keeping your son involved and making it a father/son project.
I've built all my PCs from new and used parts since 1999. I also partied, but that's another story. IMO, picking out components on a budget is the hardest part. It's a bit amazing how many pc components are mutually incompatible.
Assembling the thing is second hardest.
Almost all of my hard drive problems (not many) have come from brand new, untested (by me) drives, usually within weeks/months after formatting (I don't formally 'burn in' drives, I just bung 'em in and, after a quick SMART test, run 'em).
Used drives are the cheapest, and most easily replaced component inside a pc, unless you're using some whack job custom OS build that takes hours to reinstall and set up, because you failed to keep a config backup. Yeah, I used to do that. Anyhoo, having to reinstall software is a good thing to learn about as something to avoid. YMMV.