I read a paper yesterday, the unique choice of this form for Wigner function has to do with the properties that it must obey, so in some sense, it is the constraint that leads to the form of the Wigner function, and also by drawing the analogy between classical and quantum statistical mechanics.
The phase space density must also obey Heisenberg uncertainty principle hence the name quasiprobability distribution in the case for quantum phase space density i.e. the Wigner function as it is impossible to determine for certain the state of a single particle. Nevertheless, it must yield the distribution of q, when integrate over the conjugate variable p vice versa. It must also obey Galilean transformation, Probability distribution must be unchanged upon reflection in space and time. In the link, below one can find examples of how the wigner function applied to a simple harmonic oscillator.
DISTRIBUTION FUNCTIONS IN PHYSICS: FUNDAMENTALS by Wigner