You could get a computer capable of running Unreal 4 just fine for about 600-800$. It kind of depends on a few things.
The system requirements for UE4 are as such:
Desktop PC or Mac.
Windows 7 64-bit or Mac OS X 10.9.2 or later.
Quad-core Intel or AMD processor, 2.5 GHz or faster.
NVIDIA GeForce 470 GTX or AMD Radeon 6870 HD series card or higher.
8 GB RAM.
These are not extreme requirements, though not super cheap either. I recommend checking Kijiji/Craigslists and seeing if you can find a computer someone is selling for cheap (of course test it before you buy it if you go this route). If you want to buy brand new, you can also check out this section: Gaming & Entertainment,Top Sellers,Intel Core i5 Desktop Computers | Newegg.com-VisNav--i5_4 All of those computers will run UE4. Since it’s a big purchase though and it seems like you don’t know the technical details (I don’t mean that in a rude way), I’d recommend finding one at your price point, and asking “is this a good choice?” and people will tell you, or suggest a cheaper alternative.
Certainly you do not need to spend thousands and shouldn’t.
As for difficulty - Unreal Engine is both difficult and easy. It is a great way to learn. It’s also possible it might be too intimidating. It’s a very powerful tool. It does make things easy for the average person, but it is absolutely a commitment to get decent with it, so it’s understandable to want to make sure before you spend the cash. Especially since your son is still young. Unreal is comparable to Photoshop - tons of advanced tools, so many that sometimes MS Paint is just a lot easier to get started, even though it’s a lot more basic.
One alternative I might recommend is starting your son off with a program called GameMaker. This is kind of like Unreal, but for 2D games. It’s also a lot more simple, and can teach you the basics a lot faster. This might be disappointing in a way to your son - why start making 2D games when I can make big 3D games? But the reality is GameDev is a LOT of work, it is a challenge and takes time to learn, and the same principles will apply to both. To put it another way, GameMaker would be a good test if your son actually wants to do game dev, and wants to commit to it - it’ll also run on much older and cheaper computers.
Hope this helps in some way. It really depends on your cash flow. If you can afford 800 or so dollars without too much of a problem, it sounds like this could be a great way to get your son into Programming, which is a great field to get into, especially so young. I wouldn’t discourage it. PCs also have a lot of other value, like for school and such. Definitely do not spend more than 1200$ on PC, I wouldn’t even go that high personally, especially if you’re tepid. I’d just make sure you get at least 8 gigs of ram, and a video card with at least 4GB or VRAM (video memory) that matches or exceeds the minimum requirements.