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
The simplest method is to use HeidiSQL software. It's free and open source. It runs on Windows and on any Linux with Wine (run Windows applications on Linux, BSD, Solaris and Mac OS X).
To download HeidiSQL, goto http://www.heidisql.com/download.php.
To download Wine, goto http://www.winehq.org/.
To rename a database in HeidiSQL, just right click on the database name and select 'Edit'. Then enter a new name and press 'OK'.
It is so simple.