RENAME {DATABASE | SCHEMA} db_name TO new_db_name;
This statement was added in MySQL 5.1.7 but was found to be
dangerous and was removed in MySQL 5.1.23. It was intended to
enable upgrading pre-5.1 databases to use the encoding implemented
in 5.1 for mapping database names to database directory names (see
Section 8.2.3, “Mapping of Identifiers to File Names”). However, use of this
statement could result in loss of database contents, which is why
it was removed. Do not use RENAME DATABASE in
earlier versions in which it is present.
To perform the task of upgrading database names with the new
encoding, use ALTER DATABASE
instead (see Section 12.1.1, “db_name UPGRADE DATA DIRECTORY
NAMEALTER DATABASE Syntax”).

User Comments
This command is only available from MySQL versions >=5.1.
For older versions you can:
1. Rename all tables to be in the new database
2. Use mysqldump to dump the database data into a file and then apply it to a new database.
For more information see http://www.rndblog.com/how-to-rename-a-database-in-mysql/
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