Thread that I can pause and resume?

Conditions can be used for this.

Here's an example filling in your skeleton:

class Me(threading.Thread):
    def __init__(self):
        threading.Thread.__init__(self)
        #flag to pause thread
        self.paused = False
        # Explicitly using Lock over RLock since the use of self.paused
        # break reentrancy anyway, and I believe using Lock could allow
        # one thread to pause the worker, while another resumes; haven't
        # checked if Condition imposes additional limitations that would 
        # prevent that. In Python 2, use of Lock instead of RLock also
        # boosts performance.
        self.pause_cond = threading.Condition(threading.Lock())

    def run(self):
        while True:
            with self.pause_cond:
                while self.paused:
                    self.pause_cond.wait()

                #thread should do the thing if
                #not paused
                print 'do the thing'
            time.sleep(5)

    def pause(self):
        self.paused = True
        # If in sleep, we acquire immediately, otherwise we wait for thread
        # to release condition. In race, worker will still see self.paused
        # and begin waiting until it's set back to False
        self.pause_cond.acquire()

    #should just resume the thread
    def resume(self):
        self.paused = False
        # Notify so thread will wake after lock released
        self.pause_cond.notify()
        # Now release the lock
        self.pause_cond.release()

Hope that helps.

Use threading.Event instead of a boolean variable, and add another event for busy state:

def __init__(self):
    ...
    self.can_run = threading.Event()
    self.thing_done = threading.Event()
    self.thing_done.set()
    self.can_run.set()    

def run(self):
    while True:
        self.can_run.wait()
        try:
            self.thing_done.clear()
            print 'do the thing'
        finally:
            self.thing_done.set()

def pause(self):
    self.can_run.clear()
    self.thing_done.wait()

def resume(self):
    self.can_run.set()

edit: previous answer was wrong, I fixed it and changed variable names to be clear