if you can control MySQL, the easiest way would be to shut down MySQL, and rename the database's directory. however, i think this might mess up permissions if they're not the same on all databases.
the other way that i know of is to create the new database, dump the old one, import it into the new one, and drop the old one. say your old DB is `db1' and you want to rename it to `db2'. create `db2' and you can copy everything from `db1' with this one command:
hopefully that'll work. then you can `DROP DATABASE db1'.
mysqldump --opt -udb1_user -pdb1_pass db1 | mysql -udb2_user -pdb2_pass db2