If you are using phpMyAdmin you can go to the "operations" tab once you have selected the database you want to rename. Then go to the last section "copy database to" (or something like that), give a name, and select the options below. In this case, I guess you must select "structure and data" and "create database before copying" checkboxes and, finally, press the "go" button in that section.
By the way, I'm using phpMyAdmin in Spanish so I'm not sure what the names of the sections are in English.