How to add claims in a mock ClaimsPrincipal

First, you are missing this line in your test:

Thread.CurrentPrincipal = cp.Object;  

(and then cleaning it up in TearDown).

Second, as @trailmax mentioned, mocking principal objects is impractical. In your case, ClaimsPrincipal.FindFirst (according to decompiled source) looks into private fields of its instance, that's the reason mocking didn't help.

I prefer using two simple classes that allow me to unit test claims-based functionality:

    public class TestPrincipal : ClaimsPrincipal
    {
        public TestPrincipal(params Claim[] claims) : base(new TestIdentity(claims))
        {
        }
    }

    public class TestIdentity : ClaimsIdentity
    {
        public TestIdentity(params Claim[] claims) : base(claims)
        {
        }
    }

then your test shrinks down to:

    [Test]
    public void TestGetName()
    {
        // Arrange
        var sut = new HomeController();
        Thread.CurrentPrincipal = new TestPrincipal(new Claim("name", "John Doe"));

        // Act
        var viewresult = sut.GetName() as ContentResult;

        // Assert
        Assert.That(viewresult.Content, Is.EqualTo("John Doe"));
    }

and it now passes, I've just verified.

You don't need to mock ClaimsPrincipal it has no outside dependencies and you can created it un-mocked:

var claims = new List<Claim>() 
{ 
    new Claim(ClaimTypes.Name, "username"),
    new Claim(ClaimTypes.NameIdentifier, "userId"),
    new Claim("name", "John Doe"),
};
var identity = new ClaimsIdentity(claims, "TestAuthType");
var claimsPrincipal = new ClaimsPrincipal(identity);

And I'm not sure what you are testing here. Certainly "John Doe" will not be part of viewResult.Content because it is never been set to this.