Is there a stable orbit that an object could have around the moon?
As a generalisation of this, is there a stable orbit around any moon of any planet? And under what conditions?
For bonus points. Could such a system form naturally?
No, I don't know the answer, i'm asking the question.
Planets, Moons, Minor Planets can and do have satellites
It gets better... based on these properties of orbital dynamics Tom Van Flandern, an expert in orbital dynamics, predicted the existence of asteroids having satellites (orbiting moons). http://www.worsleyschool.net/science.../withmoon.html