You could build a new Sandy Bridge with Intel integrated graphics. But, you'll find a AMD offers more for the money. CPU's aren't as fast as Sandy Bridge, but, mom won't need that speed and AMD integrated graphics have no issues with video & HD video.
even if does start to lag, you can always add one of them htpc cards. like the 5450 n such. you can find one for about like 40 bucks or so. but you shouldnt have any problem at all with videos and 2d gaming.