This is an odd subject. As remarked chaos is a largely classical phenomena which essentially means changing the initial conditions a bit changes the final outcome a lot.
Quantum chaos would be where changes in the initial conditions of a quantum system result in a large change in the final conditions of that system. However the weird thing is that, a side from certain well known and now heavily studied systems, this does not happen. Quantum mechanics does not "like" chaos at all and it is very difficult to make any purely QM system exhibit chaotic effects.
This results in a funny situation. Even though QM contains some element of inherent probability (be it with the systems themselves or our macroscopic interactions with them, depending on your interpretation) it doesn't really contain any chaos. However the average classical system is chaotic. Choas, in practice, makes predictability more difficult than quantum indeterminism. Which means a quantum mechanical system is on average more predicable in a practical sense than a classical system.
What is causing the phenomenon that prevents us from knowing with accuracy both the position and the momentum of an electron?
Different interpretations give different answers. Most physicists would say a particle does not possess a specific momentum or position, only a probability of being found possessing some momentum or position. So there is nothing to definite to know. This is what most interpretations say. Bohr (Historical Copenhagen) would have said that "position" and "momentum" were classical concepts which mean nothing to quantum systems. However we can force a quantum system to be more like our classical world from either a momentum perspective or a position perspective, but not both at once. For some reason.
Is it just the fact that small changes in initial conditions have a relatively small effect on things that are probabilistic anyway? E.g. A tiny change in a a particles possible initial position has little effect on the possible position that the particle has at some time later. The effects of small changes are effectively absorbed by the 'smudginess' of the system. Is that correct?
When QM deals with many-body systems, i.e. thousands of particles this is the reason for the phenomena I've mentioned.
An even more fundamental reason is because QM is linear. To accommodate small changes in the initial conditions, you just add on a term to the wavefunction, a very simple thing to do. The fact that you can do this can be traced back to QM being based on linear algebra. In classical mechanics however there is no general method to determine what small changes in the initial condition will do. Of course a lot of classical systems are not chaotic because small changes in the initial conditions don't make that big a difference.
For some even stranger stuff the two following facts are very interesting. Believe it or not Quantum Mechanics likes chaos so little that: (a)Should there be a chaotic system which develops chaos on smaller and smaller scales as time develops, as soon as the chaos reaches the quantum level it gets shut off. Quantum mechanics will prevent its growth to smaller scales. (b)If chaos starts off at the quantum scale, it gets kicked up to classical scales very, very fast and the quantum mechanical level returns to its original non-chaotic state.