First collect the output of the current instance configuration. You can get the instance name stored in the SQL Server metadata.

Make sure you have backup of all the database if you are changing the production server instance name.

	sp_helpserver
	select @@servername

You can change the instance name using below query.
Default Instance

	sp_dropserver 'old_name'
        go
        sp_addserver 'new_name','local'
	go

Named Instance

	sp_dropserver 'Server Name\old_Instance_name'
        go
        sp_addserver 'ServerName\New Instance Name','local'
	go

Verify sql server instance configuration by running below queries

	sp_helpserver
	select @@servername

Restart the SQL Server Services.

	net stop MSSQLServer
	net start MSSQLServer