Does conservation of momentum mean a gyroscopic inertial thruster cannot work? There is a device called the gyroscopic inertial thruster.
Here is a Wikipedia article on it.
I believe that this device cannot possibly work and that conservation of momentum proves that it cannot work.
Am I wrong about that?
Put another way: the device is basically just a fixed double pendulum: how could there be any net momentum change in any direction after one full circle?
 A: 
I was not able to convince him that this propulsion drive cannot work due to conservation of momentum. Am I wrong about that?

No, you are not wrong.
It's clear that the engine cannot work because of momentum conservation.

It's basically just a fixed double pendulum. Why should there be any positive momentum in any direction after one full circle?

There should not.
As far as is known by physics the momentum of any closed system is conserved.
A: Given the persistence of this kind of thing, maybe determining the misconceptions underlying it are not so easy to determine and resolve.
[Given that we're talking about physics students, I guess it's fair to them to sort this out rather than ignore as one might otherwise.]
For my money, I'd ask whether within each part of the mechanism Newton's Third Law is maintained.
If so, then the total force on the system is equal to the sum of the forces on the elements of the system. 
But if the Third law holds, these must sum to zero, and hence there is no net force.
