Dell Dimension 8400
Windows XP Home Edition Version 2002 Service Pack 3
Intel Pentium 4 @ 3.40GHZ
4GB of RAM (only 3 actually recognized by OS)
PNY Verto GeForce GT220 (1024MB PCI-E version)
Dynex 400 Watt ATX Power Supply
17" flat panel monitor running 1280 x 1024
Your GT 220 will actually perform ok at that resolution(although not great.) The problem is more your processor which is pretty poor at this point and well below the minimum requirements listed for the game. Worry about possibly upgrading your video card later.
Unfortunately your processor is about the best your motherboard can handle so to upgrade you will need to replace the motherboard as well. This however can actually be done fairly cheaply, especially considering your ram is DDR2 so it will be usable with the new board. What you should get really depends on what you want to spend. On the low end an Athlon II x2 or x3 will cost you $60-70 and be more than fine. Or you can spend up to $160 for a Phenom II x4 if you want. A motherboard for these processors will be about $50-80.
jyjjy is correct on your processor, the 8400 mobo and BIOS only support P4 chips but the upgrade is not as easy as stated.
The Dell 8400 uses a proprietary case and mobo. Plus the Dell copy of windows you have may have issues running on non-Dell mobos.
basically the only thing worth bringing over to a new system would be the DVD drive since a new build would be best suited with DDR3 RAM and the Best Buy PSU will have a difficult time running most modern mid level video cards.
Ah, is it one of the BTX cases? The PSU he listed does say ATX so I was assuming it was a standard form factor.
A new case can be had fairly cheaply so that may be an option. Going for DDR3 would be nice but it would significantly increase the price of an upgrade and isn't really necessary. Also surely the hard drives and any other cards in the system can be used. Basically only the case is the problem really. Maybe the version of Windows also I guess. I don't really know enough about how Dell goes about that so I'll take your word for it.
the only difference between a DELL PSU and a ATX PSU is the DELL unit lacks an "on/off" switch in the back. So any ATX PSU fill fit in a DELL BTX case and vice versa. You can grab a DELL PSU and toss it into any ATX case and it will work just fine. In fact the DELL PSU he had was most likely better then the Best Buy unit he put in there.