Hi gadetgirl
You can copy the backup of your mom's mac mini from:
/users/(username)/Library/Application Support/MobileSync/Backup/
To her new computer to:
Windows XP: \Documents and Settings\(username)\Application Data\Apple Computer\MobileSync\Backup\
Note: To quickly access the Application Data folder, click Start, and choose Run. Type %appdata% and click OK.
Windows Vista and Windows 7: \Users\(username)\AppData\Roaming\Apple Computer\MobileSync\Backup\
Hopefully this is helpfull