The black screen thing ? does it have a log-in prompt ? If so, then you'd normally have to enter username, hit enter, then password and hit enter. Then put in the command
and see what comes up.
You should at least get some sort of error, or maybe even it might start the GUI. If it tells you something about the command not being recognised or similar, then try the command
and then enter the root password and hit enter. Then try the "startx" command and see what happens.
Doing the re-install of the card(under windows), shouldn't (in theory) have affected the linux side. It maybe that your nephew got the Radeon driver working, but didn't set the system to start in graphic mode from boot/as default. Unfortunately Radeon drivers are proprietary, and can be little swines to get working - though I understand that once they're working, they're fine (actually under linux, nvidia cards have better support and I understand that they're easier to get working).
Also, you have the added nuisance, that Mandrake 10.1 (mandrake are now known as mandriva since the connectiva buy out) is getting a little dated. The latest version is mandriva 2006, and if you obtained a boxed set version, you'd probably find that it would just be a case of re-installing it - the proprietary Radeon and Nvidia drivers don't come with the mandrake/mandriva download versions (it's a licencing thing, I understand).
p.s. If you have the install discs for the mandrake 10.1, then you also might just do a re-install over the top. If it's a download version, it should (again, in theory) be able to offer you a generic version of the driver, enough to get up and running, but to achieve the best graphic results, you'd probably have to then find out how to install the Radeon driver. There are quite a few ways round this, I'm just trying to think of the easiest ones for you, especially if you're unfamiliar with Linux distros.