I couldn’t find any way to do it in the Portal so it seems you need to use an alternative, I used
- In the Server Explorer view, expand SQL Databases.
- Right click and Open in SQL Server Object Explorer
- Login, if needed
- Right click the Server (not the database) and Click New Query
- In the Query Window type:
ALTER DATABASE [oldname] MODIFY NAME = [newname]
My refresh did not work correctly so close Visual Studio and reopen to see the result. Refresh the Azure Portal screen to see the result as well.
Rename Database on Azure - Done.