The steps are the same, as in the 2010 world.
As in Exchange 2010, installation of 2013 Mailbox role creates a new mailbox database with a name “Mailbox Database *”, where * represents a long number. Exchange administrators prefer to follow a naming convention and keep things clean.
Easiest will be to rename the default database to something that suits your naming convention. Click the Edit button and rename the database & hit the Save button.
For Exchange admins who want to delete the default database, there are couple of steps to be completed. If you just try deleting it, you will be presented with an error message that the database may contain mailboxes, arbitration mailboxes etc.
These have to be moved to a different database before we can successfully delete the default one. Running Get-Mailbox –Database dbname –Arbitration shows all the arbitration mailboxes in 2013. Move them to a different database using the command below.
Next, move any user mailboxes.
Once the moves are complete, you can delete the default database from the EAC. You will be warned that the database file will have to be deleted manually.
That’s it. Job done