TL;DR: Option 3 (way down at the bottom) with a Node 202
Because upgradability and the experience with your son seem to be important to you, I'm going to rule laptops, NUCs, and other prebuilt solutions out of hand. To be clear, this isn't to say that a laptop + evideo card or something along those lines would be a particular poor solution in terms of the end product–I might even argue better–but it seems to conflict with your mission.
With that in mind, it's time to build a SFF PC!
Your $400-600 budget is plenty to build even a complete machine, so I'll focus on a few options: 1. Kitting out the build with great, new components and a focus on an immediate future upgrade. 2. A solid new platform with money left in the budget to be spent in the future on big upgrades. 3. A "game-now" system using used components to maximize performance on a budget.
Linux support is virtually universal these days. I won't be focusing on building a computer that's open from top to bottom (if you want to avoid the IME/TPM check out this post, and read up on Coreboot/Libreboot), as it's both difficult and not really the focus of this question.
Every system here will boot from day 1. Performance will obviously be the best with the 3rd option.
We'll come back to this.
They'll all be small.
We'll come back to this a bit more later–there's plenty of reasons to go either AMD or Intel, and I'll only focus on performance–but Meltdown and Spectre specifically won't be discussed. While numbers are still coming out and nothing is final, Meltdown's hit appears to be fairly minor in gaming–syscalls is where it hits the worst. Spectre is pretty much architecture aignostic (present on both AMD and Intel chips as well as ARM) with no clear fix, so nothing can really be done there.
We'll come back to this more later.
Low power is dependent on what you throw in there, and yes, does slightly clash with light gaming. Quiet is more dependent on form factor. We'll touch on this again more.
Most computer components these days do quite well on quality. The only places we'll really be concerned with this are the case and used components.
Peripherals will be covered separately.
The thing about a case is that it's ultimately a matter of taste, and I can't decide that for you. With that, I'm going to provide a range of good cases and let you choose what you like.
That said, I'm going to recommend the Node 202 on personal preference.
Very small cases
These cases are generally described as "artisan": made by specialized companies in low volumes, with extremely good build quality, at extremely high prices.
$235 (TBD), 6.9L, 305mm video card, 5x2.5", SFX-L PSU
It's the epitome of a very small case: extremely expensive, extremely tiny, extremely good build quality (primaily powder coated steel), pretty good compatibility with a few things to keep in mind, and runs a bit hot, but generally extremely good. One big issue though: you can't buy one. They're apparently targeting Q1 2018, but for now, I can't recommend what you can't buy.
$199, 5.0L, 215mm video card, 2x2.5", special
It's the smallest case in this category, but with that come some serious compromises. video card compatibility is very limited, drive mounting is limited, and most of all, PSU compatibility is virtually nil, with only two PSUs listed: a $95 400W DC-DC and a $58 200W DC-DC. If you really value space, this is the way to go; but otherwise, I'd stay away.
$255 (TBD), 7.2L, 295mm video card, 3x2.5", SFX-L
You like shoeboxes? I like shoeboxes! Made under contract by Lian-Li of mostly aluminum, it's (IMO) the prettiest case on this list. But again, it's not for sale yet (just preorders), and I can't recommend what you can't buy.
Pretty small cases
These cases are a step up in size from the very small cases, and benefits to match. These are largely cheaper, easier to work with, have better support, and are actually available to purchase!
$195, 12.6L, 317mm video card, 3x3.5" + 3x2.5", ATX (SFX-L highly recommended)
The sort of "daddy" of small cases, the NCASE M1 is the last artisan case on this list, and one of the older ones here. It has incredible support for hardware (even a slimline optical drive!), while being spacious enough to keep things from getting too toasty.
$80 or $130 with a 450W PSU, 10.2L, 310mm video card, 2x2.5", SFX-L
It's an extremely popular choice, and a good one! Nothing stands out really except for the very reasonable price.
$85 to $130, 14L, 330mm video card, 1x3.5" + 2-4x2.5", SFX-L
All very similar with nothing too stand out; the RVZ03 trades the slimline optical slot and 3.5" mount for an extra 4x2.5" mount and RGB lighting.
$70, 12L, 330mm video card, 2x2.5", SFX-L
It's literally based on the RVZ02, but smaller, cheaper, less drive mounting, and with an optional handle!
The Silverstone SG05 and Silverstone SG06 are neat, but the 250mm video card support kills my recommendation. The Fractal Design Array R2, Lian-Li Q09, Raidmax Element, SilverStone SG07 and SG08, and Xigmatek Eris EN6305 are no longer in production, though they would make perfectly good boxes if you can find them.
Not as small cases
They're bigger with support for more things! They're also much quieter. Carrying handles are a must here, since you said that regular transport was an integral part to this build. These cases seem to have largely fallen out of favor as smaller cases take over the mITX sector, but I've put them here anyways.
$60, 26.4L, 317mm video card, 5x3.5", ATX
HardwareCanucks put it best: It's a beautiful chassis at a great price with a few minor compromises for that goal.
The Corsair 380T is a great case, but unfortunately, it no longer appears to be in production, and second hand prices for it are absurd. The Silverstone CS01B is also rather neat, but the 190mm max video card length kills any recommendation. The Cougar QBX and Fractal Design Define Nano S are beautiful and compact, but too large to not have a carrying handle and still be intended as portable IMO.
I'll be leaving out case recommendations from these options; choose whatever you think is best. If you end up needing an SFX PSU, use this list and get whatever wattage you need; if you need an ATX PSU, use this list. CPU cooler is likewise dependent on case selection, but I'll be using the stock cooler as reference, or the Cryorig C7 when not applicable. I'm allocating about $130 to these parts from the rest of these builds.
Final thoughts: I would personally go with option #3. Used components are a great way to get value on a budget.
AMD vs. Intel
To keep things simple, right now, on a $/performance basis, on the middle-high end, AMD wins in heavily multithreaded workloads, and Intel wins in single threaded workloads. Intel also wins in power consumption (though most of that will ultimately come from the GPU.) Given that gaming appears to be your intended use case, I'd recommend Intel. That said, I'm going to provide AMD builds as an option anyways.
Because this is likely going to be stuffed in a small case, I didn't bother concerning myself with looks at all. That said, if you do decide this is something important, you can of course tune the parametric filters to have the colors of your choice, and get a heatspreader if the M.2 drive is ugly.
Great, new components and a focus on an immediate future upgrade.
- The CPU is a fairly obvious choice here: the i3-8350K offers fast 4c/4t unlocked for a great price. While Z370 is a tad expensive right now, it offers room to upgrade up to 6c/12t in the future.
- Because there's no stock cooler, the Cryorig C7 stands in. As I said before, this will depend on case selection.
- Since we've got an unlocked processor, we'll want a Z series board. Wi-Fi is nice while on the go, and not much more expensive, so with a little parametric filtering, we arrive that the ASRock Z370M-ITX/ac.
- RAM here is a bit funny. Normally I recommend "get whatever's cheapest!", as RAM speed (for most workloads) doesn't make enough of an impact to justify the additional price that comes with faster memory. However, because we'll be relying on integrated graphics for a little while, I decided the fairly low bump to 2666 was worth it.
- The 850 EVO is fairly fast, and quite good value. I chose an M.2 drive as minimizing the number of cables you have to deal with in a small form factor PC is really quite nice. You're paying a bit of a premium (roughly $20), but I think it's worth it. If you don't care, feel free to step down to a SATA drive.
- RAM and storage remain the same as the Intel build.
- Ryzen's performance scales about as you'd expect as you head up the line. Though Ryzen 3s are also quad cores (even with higher clocks!), because of the CCX layout, the 1400 slots into the stack about where you'd expect. It's not beating an i3-8350K, but hey–it's Ryzen.
- Options for AM4 ITX boards–even compared to the limited options on Z370–are pretty sparse. Limiting it to ITX with Wi-Fi squeezed it down to a total of three boards. I picked the cheapest from there–however, I'm not a big fan of Gigabyte. Their products are perfectly good, but their support is awful–it took me about a year to get an RMA from them. If you're willing to do the extra $20, I'd get the ASRock AB350 Gaming-ITX/ac.
- Ryzen unfortunately lacks integrated graphics. This means we're stuck buying a GPU we don't really want, which with our budget, basically leaves the GT 1030. Don't get me wrong, it's a card with a place (read: low wattage)–but spending $75 on one isn't a great use of money IMO. I'd rather get a used card, like the HD 7870 for around $68, or even a GTX 460 for around $25..
A new platform with money left in the budget to be spent in the future on big upgrades
Intel unfortunately just doesn't have a CPU in the Coffee Lake lineup at this price point, so no build from them there. I'm not going to recommend last-gen hardware in a build for which upgradability is the main selling point either.
- The CPU has been stepped down to a 4c/4t Bulldozer APU. It's not going to provide amazing performance, but it'll do, and we can keep our board around for an upgrade.
- The board needs to support integrated graphics as well now, but we ended up with the same board anyways. ¯\_(ツ)_/¯
- Memory remains the same
- I've stepped the storage down to a 1TB hybrid drive. It's not perfect, but it'll do for a while, and make for a great secondary drive in the future after upgrading to a pure SSD for the boot drive.
A "game-now" system using used components to maximize performance on a budget.
Okay, there's no neat PCPartPicker list for this, but that's what happens with used components.
- CPU | Intel X3440 Quad Core 2.53GHz LGA 1156 | $16
- It's an overclockable 4c/8t for $16. What more can you want? Well, SATA 3 support–but this isn't going to bottleneck except in sequential reads, and even then, it won't be too bad.
- Motherboard | Gigabyte GA-H55N-USB3 | $80
- It's not the easiest to find, but it appears to be the only ITX board with eSATA, Wi-Fi, and (most importantly) USB 3.0.
Alternatively, if you've having difficulty finding that motherboard...
- CPU | Intel i5-2500k Quad Core 3.30GHz LGA 1155 | $60
- Unlocked 4c/4t for a good price
- Motherboard | One of these | $110
- Again, rather hard to find, but worth it if you can.
And if you're still having trouble...
The savings aren't as much with mITX systems as full towers, but it's still a good chunk of change for significantly better performance.
AMD's older offerings just simply aren't competitive, and stock is much lower to boot.
I'm too tired to go into this in too much depth–and I think it might warrant its own question–but the Roccat Sova MK for M/KB, and a PS4/Xbox One controller if you want a controller.
I think that's about it! I hope this helped, and I'll be happy to answer any questions you have.