How do I change the database name using MySQL?

Unfortunately, MySQL does not explicitly support that (except for dumping and reloading database again).

From http://dev.mysql.com/doc/refman/5.1/en/rename-database.html:

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 was found to be dangerous and was removed in MySQL 5.1.23. ... 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.

I don't think it's possible.

You can use mysqldump to dump the data and then create a schema with your new name and then dump the data into that new database.

Follow bellow steps:

shell> mysqldump -hlocalhost -uroot -p  database1  > dump.sql

mysql> CREATE DATABASE database2;

shell> mysql -hlocalhost -uroot -p database2 < dump.sql

If you want to drop database1 otherwise leave it.

mysql> DROP DATABASE database1;

Note : shell> denote command prompt and mysql> denote mysql prompt.