If you need to change the name of a database, you will be able to modify this within the "MySQL Databases" section of cPanel.
Once you have navigated to this page, you will be able to see the original database. In the example included below, we are using the username "tempmigr", and the database name "original", thus "tempmigr_original".
You will see the rename function on the right. If you click on this icon, it will popup to a window, which will warn you to backup your database first (a very good idea, just in-case!!), and then specify your new name.
In this case, I have called it "newname", thus "tempmigr_newname".
Once you click Proceed, it will change the name, and then refresh the page, and you will see the new name come up in the database list!
Make sure you update any scripts you have to use the new database name!