Instructions
Complete the main campaign of "Fallout 3" and wait for the final cutscene to end.
Wait for game control to be returned to you, then, head down the hallway outside the room you are in.
Turn right at the door marked "Administration" and speak with the Head Elder of the Brotherhood of Steel.
Choose the first dialogue option (the friendly option) in each interaction with the Head Elder until the dialogue concludes. This ensures that the Brotherhood of Steel becomes your friend, not your enemy. If you choose any other dialogue options, this is not guaranteed.