Instructions
Turn on your Xbox 360 and sign in to your Xbox Live account, then scroll to the "Friends" menu.
Select the "Add Friend" avatar icon and then press "A."
Type in the name of your friend&'s Gamertag (Xbox Live name) or type in a random name in the Gamertag field if you don&'t have a specific one to search for. There are millions of Xbox users, so just type in something simple (e.g. "Xboxgamer") and you are likely to find someone. Press "Done" when entered to search for the owner of the Gamertag.
Select the "Send Request" option after writing a message (optional). A friend request will now be sent to the person owning the specified Gamertag.