Honestly pretty much any new PC in the world right now would be fine for most of what it sounds like you're interested in. Is it going to be a full size cabinet? If so, there's no real need to worry about getting a small PC. I have a full-size tower inside of mine.
My cabinet is about 7-8 years old and I'm running on a 3GHz single-core intel processor... nobody uses single-core anymore so I'm sure everyone is laughing at me... but my point is that everything I'm interested in within MAME works fine... it'll have problems with 3D/polygon stuff, but I'm pretty old

so I'm interested in mostly sprite-based games. Older console emulators (atari 2600, NES, SNES, Genesis, Sega CD, Playstation 1, Commodore 64 etc.) are all fine. Most Saturn games are ok. Wii or Dreamcast games, not so much.
But man, my machine is OLD and crappy by today's standards. If that's the sort of stuff you're interested in, then I think you'd be hardpressed to find a PC today that WOULDN'T be ok. If you think you might want to play the newer consoles or polygon-based games in MAME, then you might want to get a better processor now.
I will eventually upgrade, of course, but haven't really needed to yet.