You can rename a database easily via phpMyAdmin.
We'll show you how it's done:

Step 1. Login to phpMyAdmin with your 'root' user or another user who has all privileges
Step 2. Select the database you want to rename on the left side
Step 3. Click on the 'Operations' tab on the top of the menu
Step 4: In the 'Rename database to:' field, enter the new database name
Step 5: Click on 'Go'

That's it, the database has now been renamed!

* Make sure to update the configuration of your website when the database change has been done.