Add Login NT AUTHORITY\NETWORK SERVICE to SQL Server
The account will show up as just NETWORK SERVICE when you are looking at the local accounts via the UI (I recall). I'm not sure why you see one with the hostname but that would not be a system account on the local machine.

You can always just use T-SQL to add the login:

CREATE LOGIN [NT AUTHORITY\NETWORK SERVICE] FROM WINDOWS;


Then assign it whatever permissions you would like.