As Wim said, you can't get this information reliably. For a guaranteed Windows network you can retrieve the current authenticated user (to the machine) using the server variable:
System.Web.HttpContext.Current.Request.ServerVariables("LOGON_USER")
The REMOTE_HOST variable is supposed to return the machine name, but in most cases (as he said) it will only be an IP address.
Even on a LAN, this is not an advisable security practice. This information can still be spoofed locally providing internal users the capability to masquerade as other users. If it's simply being used for identification (rather than authentication or authorization) it would probably be "good enough". If you maintained a small datatable (or the networking team) that identified each pc to its expected IP address, this information would achieve what you want.