There is something going on. Which is that certain observables are fundamentally incompatible. That means firstly that you can do an experiment for one observable or for another observable, but you can't do an experiment for both observables at the same time.
And what's worse if you did an experiment for A then one for A again and then one for B the two results for A will always agree which means after doing the A experiment the result is clearly in a state that yields a particular result for the A experiment. Remember this, doing the A experiment makes the result clearly in a state that yields a particular result for the A experiment.
However if you instead first do experiment A then do experiment B and then finally do experiment A again its different. The two A experiments can yield different results when the observables A and B are incompatible observables. The order matters. So the first A experiment left it in a state that yields a particular results for the A experiment but doing the B experiment changed it to a different state. So doing these experiments changes the state.
That's what is going on. There are states and observables and sometime doing an experiment for one observable will change the state. And that is unavoidable. And it isn't about some lack of knowledge. The state says everything we know, if we knew more we'd include that to give more or different or better information about what happens in an experiment. The knowledge we get is from the experiments but the experiments change the states and this is unavoidable once the order you do them changes the results.
The behavior of an electron (and other tiny things) is said to be probabilistic because we can't say where an election will be when we measure it, but only where it will probably be.
That's not right. We state the frequencies we will get different results in an experiment if we perform each of the various experiments. Getting certain results under a certain experiment is different than being somewhere (or having a certain momentum) and having that be passively revealed.
As I understand it, the Heisenberg uncertainty principle says the more we know about momentum the less we know about location and vice versa.
Again, no. The principle says that a state that gives a low enough spread of results for position experiments will give a higher spread of results for momentum experiments. The state tells us absolutely everything there is to know about the system, not a lesser or a greater amount, a perfect everything there is to know. It's just that some states give larger or smaller spreads for some experiments. The experiments also change the state. Both follow from the fact that these experiments are required to change the states in order to have the order you do the experiments matter.
The initial state could be the kind that changes into new position states with a wide spread (changes when we do position experiments). Or the initial state could be the kind that changes into new momentum states with a wide spread (changes when we do momentum experiments). But the state itself can't change to a small spread of both any more than a note can have a short duration and a narrow range of pitch.
Is there some property of nature that makes the behavior of an electron random,
It has to do with how you change the state. Its similar to when a single celled organism divides into two symmetrical parts. The two parts can then produce different results even though they have a common past, and the differences accumulate and build up and get larger from how they interact with the rest of the world. Its how you interact and change in response to the experiment that produces the different results. The experiment is specifically designed to have different states interact differently. Thus the original state gets separated into parts each of which interacts differently until the descendants of the original wave eventually think of themselves as the only result. And the singularity of the outcome becomes a matter of perspective, each part having the perspective it is the only part.
or does it simply appear random to us because our ability to predict its location in the future is limited by our inability to determine both momentum and location in the present?
It is wrong to think it had a position or a momentum in the present. If that worked we absolutely would have made a theory like that. It didn't. So we made a theory with states. It had a state and the state evolved. The state evolves differently based on the experiments we do and the experiments change the states. And they do so by separating the state into pieces that act differently and eventually and sometimes become separated from each other and entangled with other things so that their own perspective is that they are the singular outcome of the earlier combined state.
Or, as seems likely to me, is it simply impossible for us to know whether the behavior is random because we are limited in our ability to observe the details of what is happening
We are limited. But not by a lack of knowledge. You are limits to your own perspective based on your own state. You are limited to the results of the experiments you are connected to. And you can't blame your lack of ability to predict the future any more than that single celled organism can be blamed for not predicting whether it went north or south after it divided. If you let it do the dividing experiments many times and give it a method to accumulate the relative frequencies of its experiences then the statistical predictions will be spot on.
And the singularity of the individual experience isn't a failing for a thing that divides and has each new part be able to act like it is a whole.
If you look at the Schrödinger equation description of what happens in a measurement, the state does split, and eventually in some situations has the various parts have a perspective where it is the whole state.