Assuming you are working in the classical setting over a field like $\mathbb{C}$, I'm unaware of any elementary way to identify a natural basis in each irreducible finite dimensional module. But using creative indirect methods via quantum groups, Lusztig was able to define such canonical bases for all Lie types (and Kashiwara arrived at related methods independently). While the actual construction of such bases in individual cases like $G_2$ is not easy to implement, it is in principle "combinatorial" and is "canonical" in a strong sense. Lusztig's 1990 paper is freely available online here. See in particular 0.6 in the introduction.
P.S. As Steven points out, Littelmann's work (in many papers worth consulting) provides a new combinatorial framework for many of the standard problems in characteristic 0 about representations of semisimple Lie algebras; much of this generalizes to symmetrizable Kac-Moody algebras as well. In particular, his path method provides an algorithmic way to work out bases and characters, though the best results are obtained in type A. Other powerful methods are those of Lusztig (canonical basis) and Kashiwara (crystal basis), which are closely related to each other. All of this work probably has more theoretical than practical interest. Whatever approach is used, even in type $G_2$ the dimensions of irreducible representations (computed readily from Weyl's formula) grow so rapidly that it's questionable how far one can go in practice toward finding explicit bases of weight spaces or gaining extra insight from them.