Hi Greg,
I just finished a cocktail cabinet and built it on a budget as well. I am from Australia (all prices are $AU), so don't know if the same products are available where you are but this is what I purchased for my computer to run MAME, act as a Jukebox and everything else the cabinet is used for:
Motherboard: ASUS M4A78LT-M-LE 760G AM3 DDR3 Motherboard - cost $49
CPU: AM3 Sempron LE-145 - cost: $35
RAM: 4GB DDR3 (Transcend brand) - cost $20
Video Card: on-board - cost $0
HDD: Western Digital Blue SATA3 500GB - cost $40
Power Supply Unit: SHAW Elite-680 - $17
Total cost for all computer parts for cabinet (no monitor): $161
I have had no issues playing any of the MAME games so far, but I don't really play many post 1995 games, so no idea if they would run fine or not. I didn't see the point in getting fast anything as all it needed to do was basically run MAME, play MP3 files and I also use it as a movie player (I can tilt the monitor up to watch it like a TV.
Hope that helps.