Class with a sometimes unused property
I would do this using an interface and polymorphism to represent this:

public interface ISession
{
    void DoSomething();

    // Add your properties / method definitions
}

public class ServerSession : ISession
{
     private readonly string _serverName;

     public ServerSession(string serverName)
     {
         _serverName = serverName;
     }

     public void DoSomething()
     {
         // Some Code
     }

     // Implement interface contracts
}

public class OtherSession : ISession
{

     public void DoSomething()
     {
         // Some Code
     }

     // Implement interface contracts
}


Then you would use it something like this:

public DoSomethingWithSession(ISession session)
{
     session.DoSomething();
}

foo.DoSomethingWithSession(new ServerSession());


If there is commonality between ISession implementations, you could base class that out, then inherit from that.