I guess the "correct" way to do this is to have a protected constructor on the base class which requires the state name as a parameter.

public abstract class State
{
    protected State(string name)
    {
        if(String.IsNullOrEmpty(name))
            throw new ArgumentException("Must not be empty", "name");

        Name = name;
    }

    public string Name { get; private set; }
}

The concrete states then provide a public constructor which implicitly calls the base class constructor with the appropriate name.

public abstract class SomeState : State
{
    public SomeState() : base("The name of this state")
    {
        // ...
    }
}

Since the base class does not expose any other constructors (neither protected nor public) each inheriting class needs to go through this single constructor and thus needs to define a name.

I guess the "correct" way to do this is to have a protected constructor on the base class which requires the state name as a parameter.

public abstract class State
{
    private readonly string _name;

    protected State(string name)
    {
        if(String.IsNullOrEmpty(name))
            throw new ArgumentException("Must not be empty", "name");

        _name = name;
    }

    public string Name { get { return _name; } }
}

The concrete states then provide a public constructor which implicitly calls the base class constructor with the appropriate name.

public abstract class SomeState : State
{
    public SomeState() : base("The name of this state")
    {
        // ...
    }
}

Since the base class does not expose any other constructors (neither protected nor public) each inheriting class needs to go through this single constructor and thus needs to define a name.

Note that you don't need to provide the name when you instantiate a concrete state because its constructor takes care of that:

var someState = new SomeState(); // No need to define the name here
var name = someState.Name; // returns "The name of this state"