Can I rename an RDS instance?
Yes, you can rename an RDS instance... at your own peril.


  When you rename a DB instance, the [DNS] endpoint [hostname] for the DB instance changes, because the URL includes the name you assigned to the DB instance. 
  
  http://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/USER_RenameInstance.html


Your application code requires configuration changes to be coordinated, for this reason.

See the link above for the other implications.  This is a supported and generally safe operation, but only in your maintenance window, because there will necessarily be a brief disruption as the DNS is changed and your app code is cycled in with the new database hostname.