Is it safe to change the name of a database? I have a SQL Server database which has data for several customers/devices.  We are changing the prefix of our website (asp.net) and our application dynamically loads the proper connection string based on the first part of the URL.  Is it safe for me to change the name of my database accordingly, or should I simply put a conditional statement in the connection string logic to make the new URL point to the old database? Thanks!
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.