With regards to renaming your database, see the comments above as they are accurate.
However you still seem to be having problems getting your database out of your local environment, so let's concentrate on that first and see how you go with it.
Note that there are at least two ways you can back up/export your local database:
1). Using phpMyAdmin (Or other web-based DB utility) 2). Using the command line (see the 'mysqldump' command)
You asked about using phpMyAdmin to backup your SilverStripe database so I'll address only 1). above:
As this is specfic to your own development environment and you've not mentioned how far you've got in setting this up locally, nor any error messages you may have received, myself and others can only go so far in helping you out with this.
If you've setup phpMyAdmin as per the installation requirements, it should be installed by default and accessible at this address on your computer:
http://localhost:80/phpmyadmin
The following will also work, as port 80 is the default for most webservers:
http://localhost/phpmyadmin
If neither of these work and you receive a 404, 403 or 500 error (The screen in the browser should tell you which of these has occurred) I'm pretty confident you've not set it up correctly on your system. My suggestion is therefore to go back to the phpMyAdmin docs and re-check you've performed everything correctly as per your own environment setup (e.g. for Linux, OS X, Windows etc).
Once you're setup correctly, have a crack at the DB renaming suggestions above and coe back for more help if you need it.