Ideally then, you'd want something 7th gen Intel compatible from the box, the safest choice being some sort of 200-series chipset... (As there are faster CPUs for $60, I'm not sure I'd pay too much extra for a smaller form factor build, vice just a normal mid-tower...)
Most 7th gen stuff has been out of production a few years, but, you can probably find some leftover stuff that is still new...A quick search of a Shuttle Z270R6 Cube (case/SFX PSU, mainboard barebones) had a silly price of $400+ on Amazon, which is not worth it...
A quick search for 200 series mainboards shows dwindling supplies, and ergo, not very attractive pricing, as it makes little sense to pay $150-$170 for a mainboard to utilize a 4 year old i3 (dual core/quad thread) worth $40 when much faster CPUs are available...
Now if you can find someone selling a B250 for $80 or so, great, have at it...but, I am not seeing those prices on Amazon currently.
B460 (10th gen) boards are $110, with i3-10100 about the same cost...; these two paired would be MUCH faster than a 4 year old i3, and, likely only $40 more because not overpaying for an old out of production mainboard's markup.