During my quantum mechanics lectures and in literature I sometimes hear that "the wave function, $\Psi$, contains all information of the system". This has made me feel rather puzzled so I hope you have some good explanations of what this really means. I know that the wave function describes a state and if we are talking about electrons this state is usually a probability of finding the particle at a certain position. How is this all the information of the system? How can we even know that it contains all information? Could it be true that the wave function in fact describes the probabilities of all variables related to the system, but we usually talk about position and momentum? Can the uncertainty principle be valid for other variables than just position and momentum?
How is this all the information of the system?
We make it include all the information. For instance if the particle has spin we give the wavefunction information to describe the spin in addition to enough information to tells us the relative frequency of different position and enough information to tell as the relative frequency of every possible measurement.
You have a function that is operated on by an operator. The operator's eigenvalues are the reported results that are coupled to the measurement device and the relative squared norm of the projection onto the eigenspaces associated with the eigenvalue is the relative frequency of getting that result. The state is left in a renormalized version of its projection onto the eigenspace.
Since the operators are self adjoint and hermitian the eigenspaces ate orthogonal so the sum of the squared norms of the orthogonal projections into the eigenspaces is one. And repeated measurements give the same result.
This tells you the results and the frequencies of the different results. That's everything. And the states are the things acted on by the operators, so the states tell you everything, they tell you which operators give which results (the eigenvalues) and with what frequency (ratio of square of projection and square of original) and what happens (the state is projected).
The state did this for every operator since the operators act on the space of states.
Some people turn it all around, start with the operators and then define a state as something that acts on every operator. And sometimes people want to include mixed states too.
If some operator and state couldn't go together that would be a problem.
How can we even know that it contains all information?
If you wanted a theory that predicts the individual results then you'd want more (but it isn't clear how to get better than the frequencies since you don't have enough information to get better than statistical information).
Could it be true that the wave function in fact describes the probabilities of all variables related to the system, but we usually talk about position and momentum?
Yes and no. It isn't just a probability (or even a probability). The sizes of the projections tell you the frequency of getting various outcomes. But since you end up projecting, you also end up changing the state. This can affect later results. So it's not just a sample space and some random variables acting on it, that would not explain that the order you look at different things matters.
Can the uncertainty principle be valid for other variables than just position and momentum?
Yes. It is. For instance here is a description (by me) of the uncertainty principle for any two observables.