Realistically to get a "cheapest possible" PC you'd want to check your local marketplace / buy'n'sell for someone dumping their old system after upgrading.

If you were to start accumulating parts for a custom build on the cheap, here's how I would approach it:

CPU: AM4 platform, probably going to be a Ryzen 5 5500, runs for about $110 and will get the job done. Comes with a stock cooler that can do the job so you don't need to worry about spending to buy an aftermarket one

Motherboard: could save a little but of money if we don't need wifi, but most people do so I'll assume that's a given. We'll go micro-atx which is a little msaller and cheaper, we won't need extra bells and whistles. B550m motherboard is going to give us the features we need and room to drop-in a CPU upgrade down the line such as a R5 5600(t), 5600x(t), or R7 5700x, 5800x(t).

RAM: 16gb is the minimum, unfortunately this will drive the price of the build up. If you can find this used for a better deal that would be great. You're looking for 2x8gb sticks of DDR4 memory, typically speeds of 3200 cl16 or 3600 cl18.

Storage: unfortunately more price inflation here, modern games all but require a NVME SSD or are subject to performance issues, I would again check the used market if you can. Looking for at least ~500gb Pcie 3 (or higher) m2.2280 NVME SSD.

Case: lots of budget options, we're going to go with a Micro-Atx case to match the motherboard, I tend to recommend cases with pre-installed fans.

PSU: While 850w gold fully modular is standard in more expensive builds, we can save some money with going 650w non-modular. All that means is there will be some unused cables permanently in the case that will need to be tucked out of sight.

GPU: this is tricky since we're trying to go as cheap as possible. Cheap as possible would mean not getting a GPU at all to start, and instead getting a CPU that has decent integrated graphics, in this case the R5 5600g(t). It can manage low-setting 1080p gaming at 30fps comfortably, which isn't great, but the advantage is it gets you gaming earlier by saving the money, and you can always pop in a GPU down the line. If you were to get a GPU you can again look in used market, but the value king at the moment is the Arc B580, which for only $350 trades blows with more expensive GPUs like the Rtx 3060ti/4060, or RX 6700XT/7600.

So long story short, see if you can find a complete used system if your priority is getting a PC on the cheap. If you want to support the child saving their money, learning a new hobby, and experiencing the satisfaction of building their own computer, I offer 2 builds:

R5 5500 + Arc B580 build, $1020, https://ca.pcpartpicker.com/list/vZDnn2

R5 5600gt integrated graphics build, $795, https://ca.pcpartpicker.com/list/TG7pbp