Renaming a database from portal.azure.com is the feature that was missing from the very beginning of the cloud service.
The task can be accomplished in few seconds, with a single line of TSQL if you open your database with SQL Management Studio or, even more quickly, if you use the "Query editor" blade from the portal.
This is the line of TSQL code you have to run in the command line:
ALTER DATABASE [servername/databasename] MODIFY NAME = [servername/newdatabasename]
And... boom! After few seconds your database will be renamed.
There's nothing more simple :)