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
I think the solution is simpler and was suggested by some developers. phpMyAdmin has an operation for this.
From phpMyAdmin, select the database you want to select. In the tabs there's one called Operations, go to the rename section. That's all.
It does, as many suggested, create a new database with the new name, dump all tables of the old database into the new database and drop the old database.
