i don’t know why i never answered this age-old question, sorry
okay, a couple of things. the commodore 64′s mainboard/etc were built into the keyboard, so when you say “commodore 64 case” i am assuming you’re talking about the keyboard:
if you’re talking about the 1541, i’m wasting my time. your hardest problem is going to be re-working the circuitry of the keyboard to work with your new computer inside of it. this circuitry was specific to the c64 hardware but i’d bet it follows some PS/2 paradigm somewhere along the line. if you can somehow hack it to present a normal PS/2 interface (extremely hard) you can simple use a PS/2 to USB adapter to connect it to your new machine
on to the machine. any computer that could possibly work for what you’re trying to do is going to be a 32 or 64 bit machine, so there’s that. terms like ‘minicomputer’ & ‘microcomputer’ don’t have much relevance in 21st century, so you’re not looking for anything like that
you’re not going to be able to fit any kind of modern motherboard in there, and you’re certainly not going to be able to cool it properly if you did. even micro atx is way bigger than what you’re working with. you need to go with a SoC & board
this is where it gets kind of screwy. you’re now in the realm of “embedded” computing and it follows you should use some ARM chip (don’t use the intel edison or whatever, there’s probably an x86 box like the old eeepcs that might do what you’re looking for but i have no idea about embedded x86 stuff).
if you’re going cheap, the beaglebone black is always a good option. for cutting edge ARM64 stuff, there’s this:
i’d go with the hi-key, the snapdragon one is nonstandard ARM chip that qualcomm makes for phones. you’re probably going to run into problems there as qualcomm would sooner kill themselves than give you any kind of documentation