PseudoBrain FizzBuzz Thoughts
I'd first create an interface that your FizzBuzzConverter class can implement:

public interface IConverter
{
    string Convert(int value);
}

public class FizzBuzzConverter : IConverter
{
    // ...
}


Then the PseudoBrain can have any implementation of the interface injected into it:

public class PseudoBrain
{
    private readonly IConverter _converter;

    public PseudoBrain(IConverter converter)
    {
        _converter = converter;
    }

    // ...
}


This will allow for easier unit testing via mocking or writing and using new Converters.

Lastly, I have a bit of a squicky feeling when I see random number generators created over and over (as Think() will be called repeatedly). So I put that at the class level:

public class PseudoBrain
{
    private readonly Random _random = new Random();
    private readonly IConverter _converter;

    public PseudoBrain(IConverter converter)
    {
        _converter = converter;
    }

    public virtual string Think(string lastResult, int value)
    {
        var ai = _random.Next(0, 99);

        // ...
    }

    // ...
}