Hello,
I assume you will be running AFNI under Linux.
If that's the case, then the minimum would be pretty hard to get these days, you can easily run afni on a 500Mhz computer with 256 Mb of RAM.
HOWEVER, you're better off trying to get the following
Memory: 1 Gb
CPUs: whatever is out there these days. If you plan on processing very large functional data sets with 3dDeconvolve and what your results quick, a multi-processor machine will speed up your analysis. But in general you'll be OK with one processor if you don't want to spend the money on it.
Video Cards: The ATI is OK for AFNI, but if you want to use SUMA then I recommend you get the Nvidia graphics cards because they have good drivers for linux. We have the Nvidia GeForce 3 and 4 on our systems and they both work very well.
cheers,
-z