You would need to do this then, won't affect anything else unless you have any jobs that use the old server name. If so you need to updated them with the new server name.

1. sp_dropserver 'old_name'

2. go

3. sp_addserver 'new_name','local'

4. go

Also restart sql server as

net stop mssqlserver

net start mssqlserver

This should do the trick for you

Thanks!!