In QM, does random data "come from anywhere"? Also, what are the properties of the data? I have only taken a basic quantum mechanics course (this book, so you know where I'm coming from), but I've been wondering about something.
If we set up a quantum system in a known state and take a measurement of two incompatible observables, we will get two real numbers.  If we repeat this experiment multiple times, then we will get two lists of real numbers (each list corresponding to the measurements of one of the observables). Quantum mechanics allows us to predict the average and standard deviation of these numbers, but it does not allow us to predict the individual numbers.
If I understand correctly, this is a fundamental limit of the theory.  The data is essentially random.  Is it correct to say that most scientists believe that no theory will ever allow the prediction of these individual numbers?  Why do they think that?
And secondly, is there any other property of those numbers that quantum mechanics predicts that I am missing (other than mean and standard deviation)?
 A: The random data are genuinely random. This is actually a question that quantum mechanical theories have the potential to answer as well. One may ask what is the probability that any pattern (or correlation with something previously uncorrelated) emerges in the random data and quantum mechanics allows one to calculate that the probability is "zero".

If I understand correctly, this is a fundamental limit of the theory.

You don't understand it correctly. The random character of the outcomes of measurements is a fundamental property of Nature that is reflected in the correct theory but because the theory reflects something that is exactly true, it doesn't follow that this feature is any sign of incompleteness of the theory. The theory is totally complete. The random character of predictions doesn't show any limitation of the theory; it shows an important fact about the natural phenomena.

The data is essentially random.

No, the data is exactly, not just essentially, random – randomly chosen according to probability distributions that may be calculated.

Is it correct to say that most scientists believe that no theory will ever allow the prediction of these individual numbers?

Truth in science isn't searched by counting "most scientists". Most scientists, if one includes biologists or perhaps much less "hard science groups", may misunderstand quantum mechanics, but it changes nothing about the facts. The fact about the randomness of quantum mechanics predictions is known to every competent physicist because it's been a well-known insight and textbook material for over 80 years.

Why do they think that?

Because it unambiguously follows from the most thoroughly and accurately tested theory in the history of science. And because any mechanism that would try to deterministically assign the results of the "random data" would violate the basic postulates of quantum mechanics. It would have to become a completely non-quantum "meta-theory" and this "meta-theory" would also demonstrably violate other verified principles such as the laws of special relativity.
