In all discussions regarding the Lagrangian formulation it has always been said that $L = T - V $, only is a correct guess that when operated via through the Euler -Lagrange equation yields something similar to the newtons law of motion. But if we are to consider yourselves at the position of Lagrange himself, how are we to guess the correct form of the Lagrangian, using physical arguments only. Even considering that nature is always trying to minimizing something, How do we say it is the action that it must, not something else?
Generally I like to think of it as a generalization of the concept of static equilibrium, so that it becomes a "dynamical equilibrium". Here are some lecture notes that i found to be quite helpful in grasping the meaning and origin of the subject. http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf