You are probably trying to CREATE a NEW USER with the same name. All you need do is associate the new database with the EXISTING user. (The existing user for the original database).

If you are using proper C-Panel, scroll down to the bottom of the MySQL manager screen. There you will see a dropdown with the existing user and another dropdown with the databases. Your NEW database should appear. Simply SELECT the existing user, and associate that user with the NEW database. You need to give ALL PRIVILEGES to the user, for the new database.