CD/DVD Burner
http://www.newegg.com/Product/Produc...82E16827136119
Case w/450W PS
http://www.newegg.com/Product/Produc...82E16811147031
160GB HD
http://www.newegg.com/Product/Produc...82E16822136075
17" LCD Monitor
http://www.newegg.com/Product/Produc...82E16824254018
Keyboard (I have this, it's AWESOME!)
http://www.newegg.com/Product/Produc...82E16823107120
Microsoft Mouse
http://www.newegg.com/Product/Produc...82E16826105026
1GB Ram
http://www.newegg.com/Product/Produc...82E16820146580
MSI Motherboard w/ Intel Graphics (Good enough for vids, movies, basic stuff)
http://www.newegg.com/Product/Produc...82E16813130060
Dual Core Core2Duo 2.0GHz (Very fast CPU)
http://www.newegg.com/Product/Produc...82E16819115014
Vista Home Basic
http://www.newegg.com/Product/Produc...82E16832116196
$619.90 for a very nice PC, including everything. Take out the monitor and you can add another gig of ram and faster core 2 duo, or take off $149.99 and your total is only $469.91, under budget and will definitely last a few years.
Since Core2Duo's are very cheap, yet some of the most powerful CPU's currently out (RIP AMD

), you should not consider anything less.
I assume your grandma doesn't play games, so a dedicated card is not needed. The motherboard is certified for Vista, reviews online say it can handle the Aero interface just fine, which will probably be of little concern anyways. Even low end integrated graphics chips will handle movies and videos just fine.