Uh, MAME? That's about all you've got to work with there.... I suppose you could pick up one of the 1xxx-in-1 boards, but it's just a PC with a JAMMA I/O board slapped on to it... plus it's underpowered and many times the components you get are used garbage.
Just get the smallest motherboard you can find and build a MAME PC with it.
To be clear though ... there is NO single board user programmable option out there other than the Arcadeshop board which has a limited number of game, an absurdly high price, and a guy running the show that doesn't seem to care about after the purchase customer support (no updates since JANUARY!).