-- Get a list of all the  remote logins you will need to disable:
select 
    sl.remote_name as RemoteLoginName, 
    ss.srvname
from sys.remote_logins sl 
inner join sys.sysservers ss on 
    sl.server_id = ss.srvid

-- Be sure to record the remote logins so you can recreate them afterwards.
-- To drop any remote logins you may be found, you can use the following syntax:
-- For a default instance
sp_dropremotelogin <old_physical_server_name>
GO
-- For a named instance
sp_dropremotelogin '<old_physical_server_name\instancename>'
GO

-- Once the remote logins are out, you can go ahead and change the name of the server with this script:
-- verify name incorrect
SELECT @@servername;

-- Remove the old name from the SQL server
sp_dropserver '<name from @@servername>';
go

-- Add the new name, with local set as well.
sp_addserver '<new server name>',LOCAL;
GO

-- verify change taken
SELECT @@servername;

-- To add the remote logins back after you have changed the server name you can use the following syntax:

sp_addremotelogin 
    @remoteserver = '<remoteserver>', 
    @loginame = ] '<login>', 
    @remotename = '<remote_name>'