In whatever SQL designer you are using Create a new database and there should be a command to copy your old database schema with tables to the new database with the new name.
For a brief time mysql had the ability to do that. They removed it because it was deemed to be dangerous and peoples data was becoming corrupt.
exporting the database, creating a new database, and importing the old data into the new db is a good option, and it will work without problems.
Another (possible) option that seems to work for innodb db/tables, is to
create a new database, let's say "new_db"
for each existing table, run RENAME TABLE old_db.table TO new_db.table; where old_db is your original database name and new_db is the new one, and table is obviously the individual table name
I haven't tried that, but it's supposed to be pretty damn fast.