Why does it appear that my random number generator isn't random in C#?

The problem with min = 0 and max = 1 is that min is inclusive and max is exclusive. So the only possible value for that combination is 0.

random = new Random();

This initiates random number generator with current time (in sec). When you call your function many times before system clock changed, the random number generator is initiated with the same value so it returns same sequence of values.

Don't create a wrapper method for Next. It wastes cycles creating a new instance of the Random class. Just use the same one!

Random myRand = new Random();

for(int i = 0; i < 10; i++)
{
    Console.WriteLine(myRand.Next(0, 10).ToString());
}

That should give you ten random values.

As has been said--Random is pseudo-random (as all implementations are), and if you create 100 instances with the same seed, you'll get 100 instances of the same results. Make sure that you're reusing the class.

Also, as folks have said, beware that MinValue is inclusive and MaxValue is exclusive. For what you want, do myRand.Next(0, 2).