Is it safe to change the name of a database?
If you have queries that reference the full database name (which you likely will have if you have cross database queries), they will break. If you have Imports or reports that reference the full database name those will break. If you have linked servers (SQL server), those will break.  If you have other applications that hit the same database backend, those will likely break.

If you have none of those situations, you may be OK. However, you will still want to do this only if you have a full test suite to be able to verify that nothing broke. Keeping the database name and changing how the application connects is a far safer choice if you don't have a good test suite.