One of my clients asked me to be able to take manual backups of his DB, and because he wasn't so technologically-inclined, using the hosting company's interface wasn't an option. So I created a small PHP page that uses mysqldump to dump the site's db in a downloadable way. The interesting part is:

// Make sure user is allowed to dump the DB. We don't want everyone getting a copy of our data!
if (UserIsAdmin()) {
 // Replace variables with your own values
 $command = "mysqldump --user $user --password=$password --host $host".
 " --create-options $db".
 $arr = null;
 // Execute
 $res = implode('
', $arr);
 // Make the resulting input look like a downloadable file. Remove these if you wish
 header('Content-Type: text/plain; charset=utf-8');
 header('Content-Disposition: attachment;filename="backup.sql"');
 echo $res; 

And that's it!
Of course this script only works if you can execute stuff on your host.
You can give it its own page, or make it a part of a bigger one.