Xpendable:
I doubt he needs anything more than an UNO. Did you look at the photo of the machine? There is no CRT or graphics. It is a mechanical game. The code would probably be less than 100 lines and a handful of variables. No need for 4 times the SRAM.
Newbies in coding have a difficult time with efficiency. When s/he comes back and ask for suggestions on logic and state, one of the elder members will surely suggest a large array. Now down the road of SRAM consumption we go...
The different in price for the Chinese clones is only 2::1 in the regular Arduino formfactor, that is under $10 for UNO-ish and under $20 for Mega-ish. $10 is a cheap insurance policy!
Ray