I'll echo Ben Webster's comment from the other thread, but in the other direction: find some natural commuting operators whose simultaneous eigenspaces are all one-dimensional.  This is how, for example, one gets the root space decomposition in Lie theory and the decomposition of spaces of modular forms into eigenforms.  (Admittedly, the former is only unique up to a choice of Cartan subalgebra.)

I guess that to get an actual basis instead of a basis-up-to-scalar-multiplication one needs a little more.  For cusp forms one usually takes the normalization where the coefficient of $q$ in the Fourier expansion is $1$, whereas for the Lie algebra I guess we take a normalization where the structure constants are as nice as possible.