They definitely could have installed a new mobo, cpu, etc in her Dell case and you could as well but you will need to get a new mobo, Processor, and memory because the memory you have will not work will a new Intel or AMD mobo. Given the machine is 3 to 4 years old, it would be advisable to get a new hard drive, although you could use the old one. Again, if it's only 5400 RPM, it will definitely slow things down. You could also use your old CD ROM drive but given a new DVD Burner would cost about $30, you could get a new one. Basically, replacing the mobo, CPU, and memory is pretty much building a new computer in that these are the high cost items. Unless you get an onboard graphics card you'll need a new one because the one you have in the Dell will not fit a new mobo. You can get a nice case for about $50 or really good one for about $100, so your not really adding a huge cost getting a new case. Here are examples of cases both of which come with good Power Supplies although the second is better.
http://www.newegg.com/Product/Product.aspx?Item=N82E16811147054
http://www.newegg.com/Product/Product.aspx?Item=N82E16811129155
My thing with kids and computers is that I don't want them using mine, kids are virus/spyware magnets.
What I do is give them my old one and build a new one for myself.
Just getting a new drive and increasing the memory on the Dell would make it better for games. With games especially, the graphic card is also important but you're looking at $100 or more for a decent graphic card.
I know you said you wanted Intel but AMD prices are dropping big time and there should be another drop in a couple of weeks. Regardless of whether you want to use the Dell case or get a new one, what you get will depend on your budget and how you use the machine, e.g,. do you have a camcorder, will you overclock, what kind of games do you play? If you give us your budget and how you will use the machine, we can make some suggestions. If you do decide you want to use the Dell box, measure the mobo, length and width. A Mciro ATX will be about 9.6 by 8.6, an ATX will be about 12.0 by 8.6 or so. As for putting it together, there would really be no difference whether you used the Dell case or a new one, the only advantge of a new one is that the documentation might be better. Putting a computer together is very easy and the instructions that come with most mobo are generally very good. Plus, if you put it together yourself, your geek cred with your son and his friends will go way up.