Particle, you've got a few options.
First, if you look though the Dell discs for your mom's PC, you might find one that has 'Drivers and utilities'. When you load this disc, it'll have a bunch of available drivers. If it's found specific hardware on your machine, it'll put a checkmark next to it in the list.
If you don't have the disc, you'll need to go to
http://support.dell.com from another PC. Once there, go to the 'Drivers and downloads' section, and enter the relevant information about your PC. This will take you to a page full of drivers.
You want to go to the Network section and download the ones that say they're Drivers and not Utilities. For a Dimension 8300, they have the following network drivers:
Broadcom - Driver
Applies to:
NetXtreme Gigabit Ethernet
and
Intel - Driver
Applies to:
10/100 LOM
I have no idea which you'd have in your PC. My guess it, though, that if your NIC is onboard (meaning that your ethernet port on the PC is directly attached to the motherboard, and it's not on a card in one of your PCI slots) you'll have the Intel drivers, and if your NIC is a PCI card it'll be the broadcom driver.
Good luck.