It depends on the games you want to play. For example, if you like early 80s games you can get buy under 500MHZ. If you like 16-bit games of the late 80s, get a P3 of at least 800MHZ. If you like 32 bit games, then you need a fast PC of 2GHZ or more.
As for RAM, you can run most games with 128MB. However, 256MB is needed for some newer Neo-Geo Games, and some other very large games.
For a cheap PC, you may want to check local stores that deal in generic brand equipment. You can get a lot of PC for a lot less, buying local, with generic brand componets. If this is going to be a MAME-Only PC, you can leave out printers, DVD drives, high end keyboards, etc to save cost.