The MySQL manual at MySQL covers this.
Usually I just dump the database and reimport it with a new name. This is not an option for very big databases. Apparently
ALTER DATABASE is the proposed way around this by MySQL and RENAME DATABASE is dropped.
RENAME DATABASE
From 13.1.32 RENAME DATABASE Syntax:
RENAME {DATABASE | SCHEMA} db_name TO new_db_name;
This statement was added in MySQL 5.1.7, but it was found to be dangerous and was removed in MySQL 5.1.23.