How to pause/suspend a thread then continue it?

You should do this via a ManualResetEvent.

ManualResetEvent mre = new ManualResetEvent();
mre.WaitOne();  // This will wait

On another thread, obviously you'll need a reference to the mre

mre.Set(); // Tells the other thread to go again

A full example which will print some text, wait for another thread to do something and then resume:

class Program
{
    private static ManualResetEvent mre = new ManualResetEvent(false);

    static void Main(string[] args)
    {
        Thread t = new Thread(new ThreadStart(SleepAndSet));
        t.Start();

        Console.WriteLine("Waiting");
        mre.WaitOne();
        Console.WriteLine("Resuming");
    }

    public static void SleepAndSet()
    {
        Thread.Sleep(2000);
        mre.Set();
    }
}

var mrse = new ManualResetEvent(false);

public void Run() 
{ 
    while (true) 
    { 
        mrse.WaitOne();
        printMessageOnGui("Hey"); 
        Thread.Sleep(2000); . . 
    } 
}

public void Resume() => mrse.Set();
public void Pause() => mrse.Reset();