While standard solar model (SSM) predictions depend on approximately 20 input parameters, SSM neutrino flux predictions are strongly correlated with a single model output parameter, the core temperature T-c. Consequently, one can extract physics from solar neutrino flux measurements while minimizing the consequences of SSM uncertainties, by studying flux ratios with appropriate power-law weightings tuned to cancel this T-c dependence. We reexamine an idea for constraining the primordial C + N content of the solar core from a ratio of CN-cycle O-15 to pp-chain B-8 neutrino fluxes, showing that non-nuclear SSM uncertainties in the ratio are small and effectively governed by a single parameter, the diffusion coefficient. We point out that measurements of both CN-I cycle neutrino branches-O-15 and N-13 beta-decay-could, in principle, lead to separate determinations of the core C and N abundances, due to out-of-equilibrium CN-cycle burning in the cooler outer layers of the solar core. Finally, we show that the strategy of constructing ¿minimum uncertainty¿ neutrino flux ratios can also test other properties of the SSM. In particular, we demonstrate that a weighted ratio of Be-7 and B-8 fluxes constrains a product of S-factors to the same precision currently possible with laboratory data.