Graphics card question for the mini

I want to be able to watch my college streaming HD broadcasts on my TV. My iMac is too old and I can't seem to do it on an Apple TV. The only way I can figure out to do it is thru a browser on a Apple Mini.

If I got a mini to do this and hooked it directly from the HDMI output to the HDMI input on the TV, how much impact would the graphics card have on it? I would lean toward the better graphics card & memory for $200 plus a 7200 rpm hard drive, but would that really useful in a media center type application? I know I will have to upgrade the memory and I can do that myself(it's so easy now). The last thing I want to do is buy all this stuff and have it be super slow.

Thanks in advance