Sattelites of the moon are unstable: too close and it notices the lumpiness and does not make a nice ellipse, so after a while will crash. If too far, it will fall out and orbit the Earth instead.
Look at the “Hill Sphere” represented as a topological map, and you get the feeling for how the gravity of the Earth interferes, leaving only a small region where orbits look like they would go around in a normal manner.
But you can’t orbit that close, since the moon is not just lumpy but off ballance. As it turns out, certain high-inclination orbits work out better, on paper anyway.
So first, you need to find out what makes the hill sphere bigger. I don’t know off hand, but I suppose the moon can be larger and farther away. But if the moon is far from the primary, it has the same issue and will end up orbiting the sun, or easily peturbed from other bodies in the solar system.
Also, our moon is rather lopsided. A more suitable situation would be to have the moon be especially symmetric. Perhaps it could be liquid? That would allow the sattelite to orbit close in and remain stable.
Throw in some other reason to remain stable, like the all-powerful resonance. Make it slightly eccentric, make the sub-sattelite off-ballance and in an odd half multiple of its period, which is also a small multiple of the moon’s period, and furthermore have another large moon with a period an exact integral multiple of the moon’s.
If that’s not quite right, it’s at least believable to the audience, if they don’t solve the math for real.