What is the exact meaning of the wave function of a particle?
I can certainly understand your wish to better understand the rather strange nature of wave functions. But I agree with some of the comments above that we try to steer away from questions like "what is the meaning of . . ." because they start all kinds of nasty debates about what the meaning of meaning itself is and so forth. It might help to focus on the question like this:
How do we arrive at a wave function and what does a wave function tell us about the particle or system?
I agree with Charlie above - a wave function can be used to further deduce most everything you would want to know about a particle or system. We can get the momentum of a particle through the use of momentum operators, or the potential energy of a particle at a particular position, or the angular momentum through the use of angular momentum operators and so forth.
Lastly, non-relativistic wave functions will not tell us the mass of the particle or system. You have to know the mass of the particle beforehand. Ordinary quantum mechanics (as studied in undergrad quantum) doesn't tell us the spin, although spin-friendly versions of wave functions do exist.