No, not really.
A "database name" on the web is actually the name of a database server which allows remote connections to it. In order to use a "random name" you would have to continually change the name of the DB server instance to match it and that's not a trivial task.
A better idea would be to set up a
web service[
^] which handles the DB for you, and which your app connects to using your "random key"
I'm not at all sure what you think you would gain from doing this though - if you can access via any random key, then so can I - so it doesn't enhance security in any meaningful way.