Rename MySQL database

It says here that:

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 9.2.3, “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.

You can import all data into a new database then delete the old one.

It is also possible to use the RENAME TABLE statement using same from/to table name but different from/to database. There are a few catches.

CREATE DATABASE `test_bkp`;

RENAME TABLE 
`test`.`table1` TO `test_bkp`.`table1`,
`test`.`table2` TO `test_bkp`.`table2`,
`test`.`table3` TO `test_bkp`.`table3`;

In phpmyadmin you can just click on your database, then go to the Operations tab which lets you rename it.