As far as I understand, in classical (not quantum) physics the equations of motion are established experimentally (e.g. Newton's three laws and Newton's gravity law, Maxwell's equations). Then sometimes it turns out to be convenient to rewrite them as Euler-Lagrange equations for suitably chosen Lagrangian $L$. However $L$ is not determined uniquely by the equations of motions, and may be chosen in any convenient way without affecting the physical results.
The situation changes when one considers the quantum version of the equations. Usually one has to choose a particular form of $L$ and either substitute it to the path integral, or construct a Hamiltonian out of it, guess a quantum version of the Hamiltonian, and substitute into the Schroedinger equation, sometimes using further analogies (e.g. so it worked with the Pauli equation, I think).
In all the literature I know it is never explained how to choose the right $L$ for quantizing a problem. All textbooks just ignore this step. I realize that the final criterion of correctness of a theory is its consistency with experiment, but still I am wondering whether there are more theoretical arguments to choose a specific $L$?
