Mine is this the VIA EPIA SN:
http://mini-itx.com/store/?c=2#epiasnI bought this one almost solely for the 4 SATA ports, as I use it to build a Network Attached Storage PC.
If you're getting a VIA board, you might even consider sacrificing some CPU speed for a fanless setup (which will probably still need a case fan as I'm not sure about the circulation in a NES case) which would make it completely quiet.
I don't know what the performance of the VIA processors are compared to an Intel Atom 1.6, as Atom hadn't hit the stores when I bought mine, and power consumption was more important for a PC running 24/7.
Also note that you'll need a small power supply, either internal or external. I use an internal 80W PSU bought from mini-itx.com. And depending on how much space is in the NES box, you might need a a 2.5" harddrive. (Any normal SATA drive for laptops will do)
Something that many people miss when building stuff like this is heat dissipation. Before sure you test it thoughtly by running intensive tests for a few hours in an environment similar to where the box is going to be placed. If it runs too hot, start drilling holes for fans unless the NES case has some already.
If you're really awesome, you'll solder on the on-switch to the one on the NES, and possible add a DVD-drive where the cases usually go. Then again, physical media like DVDs might not even be necessary.
All in all, it sounds like a freakkin' cool project and one hell of a gift. If my girlfriend gave me this, I'd marry her on the spot