If it's an old laptop your best bet is to start over anyway. It's probably got a ton of stuff you don't want/need and it'll run best after a fresh install. There may be a recovery partition that you can use to reset it to factory defaults. Google for instructions on how to do this for the specific model, something like, "reset HP XYZ to factory." You probably need to hold down a couple of keys on startup to get to that part.
Now, if your mom has any important files on it you can download a linux live cd (you don't have to install the operating system to use it) and copy them that way. If you still want to just reset the password you can also do that with the live cd. Instructions are here.
edit: Also, Windows 8 consumer preview is free to download and will probably work at least through the end of the year. That's another option, especially if you don't have a disc or recovery partition. If that computer has Vista on it Win 8 should run fine, and probably even better.
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