Your pics didn't show up... But where are you doing this from??? If you're in phpMyAdmin it should be fine... If worse comes to worse you can always export the data, drop the table, recreate it and import the data you extracted.
That might actually be what he'll have to do. Read below for why he might have to do it that way.
Curiosity though: How many SQL databases are you allowed to have?
Reason for asking: PHPMyAdmin doesn't do renames. It recreates the database with a new name, gives it the same structure, puts the data in it, then deletes the old one.
If you're only allowed to have 1 database (which you might), or you're already at the maximum number of databases you're allowed to have (which you might), renaming with phpmyadmin won't work because it needs to create a new one to do the rename.
Not sure _why_ it has to create to rename, but I didn't build it - I just know it'll have 2 databases with the same data for a moment or two while it moves data from the old database to the new one with the new name.