ALTER DATABASE "DATABASE-OLD" --OLD
SET SINGLE_USER WITH ROLLBACK IMMEDIATE
EXEC rdsadmin.dbo.rds_modify_db_name N'DATABASE-OLD', N'DATABASE' --this is similar to on-prem "ALTER DATABASE dbName MODIFY NAME = dbNewName"
ALTER DATABASE "DATABASE" --NEW
SET MULTI_USER WITH ROLLBACK IMMEDIATE
To delete a database that has been set as Offline, you first have to bring it back online, and then drop it, like so:
EXEC rdsadmin.dbo.rds_set_database_online N'YOUR-DATABASE-HERE'
EXECUTE msdb.dbo.rds_drop_database N'YOUR-DATABASE-HERE'
If you have problems, you may try renaming the database and trying again:
EXEC rdsadmin.dbo.rds_modify_db_name N'YOUR-DATABASE-HERE', N'DeleteMe'
If none of these work, you may not have the correct permissions, which could require resetting your master password — see the AWS documentation for that.