Purely formal considerations suggest that the Lie bracket of quantum observables should be proportional to the Poisson bracket of their classical counterparts. (In particular, the classical Poisson bracket satisfies$ \lbrace u,wx\rbrace v +u \lbrace v,wx\rbrace
=\lbrace uv,w\rbrace x + w \lbrace uv,x\rbrace$, and the proportionality follows from this and a little algebraic manipulation (after dropping the commutativity assumption.))
Classically, the Poisson bracket of position and momentum is a constant (namely $ 1$), so in the quantum setting we want the Lie bracket of position and momentum to be a constant (in this case purely imaginary so that $p$, $q$ and $[p,q]$ can all be Hermitian). That is, we want to identify position and momentum with Hermitian operators satisfying $[p,q]=i\hbar$ for some real constant $\hbar$.
The usual choices satisfy this condition, and are particularly simple, so they get used. Other choices would work just as well in principle but might be a little messier to manipulate. This is all explained quite clearly in Dirac's book on Quantum Mechanics.