To begin, is the Mac Mini getting internet from a wired router? (through ethernet or USB) If this is the case, you would be better of sharing the mac mini's connection over airport than over bluetooth as the range and speed will be much better.
In system preferences, go into sharing and select internet sharing from the list on the left. Before you tic the box, chose share your connection from: ethernet (I guess) and to: Airport. Under AirPort Options (the button on the bottom left) you should select a type of encryption (WPA is recommended over WEP) and enter a password.
Once that is done, your Girlfriend can connect to the Mac Mini as if it were a normal wifi hot spot.
