This is my "budget gaming computer":
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It's a standard Dell desktop, not one of their specialized gaming systems. But the CPU is water-cooled, and I sprung for a higher-end graphics card. It's fast. It runs quietly, except when I'm playing anything with modern high-end graphics. Then its fans start howling like jet engines. Total cost ~$2k.
In my mind, this represents probably the minimum viable configuration for a modern gaming PC. I paid a premium to have the vendor source and assemble all the components for me, rather than spend my time tracking down discounts on each of them. If your discount-fu is strong, and your time is free, you can probably get something close to this for less than I paid.
However, I'm not sure building a budget gaming PC for someone else is a great idea. If it's not up to snuff, then it's just a white elephant they can't actually use but have to pretend to appreciate because it's a gift. The only way I see it working is if you were in a PC repair or assembly business, and had a lot of high-end components lying around at wholesale or used prices. Then you could consider eating the parts cost and donating your labor, as a gift, but still give your son a high-end gaming system.
I think going the budget DIY route is much more satisfying when it's your own project for yourself. When you know how much PC you want, and you know whether you'll be satisfied with however much PC you can assemble on your budget. Since this is for someone else, I think a better gift would be to work with them to build a PC that fits their specs and their budget - gifting some parts from your budget to round out or level up the build.