Unfortunately, the ArcSDE geodatabase cannot be renamed in SQL Server.
This also applies to moving or copying an ArcSDE geodatabase from one server to another using SQL Tools such as Detach and Attach, Backup and Restore, or the Copy Database wizard.
No matter which of the methods you use to move your SQL Server database, you cannot rename the database. When you are restoring a database, for example, you are given the opportunity to restore it with a different name. Don’t do this with a geodatabase; you won’t be able to connect to it.
All object names in the geodatabase system tables are fully qualified with the database name. In addition, many stored procedures use a three-part naming syntax in their code, which follows the format <database>.<owner>.<object>. If the database name changes, you will not be able to execute these procedures.
This is further reiterated in the ArcGIS Resource documentation referenced from: http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#//002q00000087000000