Having successfully taken a database instance offline using the Management Studio I then attempted it bring that same database back online both using the Management Studio and by executing the following script.
ALTER DATABASE [dbname] SET ONLINE
This results in the following error
User does not have permission to alter database ‘dbname’, the database does not exist, or the database is not in a state that allows access checks.
Turns out that on RDS you can use the standard SQL script above to take an instance offline, but Amazon require you to execute a stored procedure to bring it back online;
See here for more http://docs.amazonwebservices.com/AmazonRDS/latest/UserGuide/Appendix.SQLServer.CommonDBATasks.html