Background: Parity-violating electron scattering provides a model-independent determination of the nuclear weak-charge form factor that has widespread implications across such diverse areas as fundamental symmetries, nuclear structure, heavy-ion collisions, and neutron-star structure. Purpose: We assess the impact of precise measurements of the weak-charge form factor of 48Ca and 208Pb on a variety of nuclear observables, such as the neutron skin and the electric-dipole polarizability. Methods: We use the nuclear density functional theory with several accurately calibrated nonrelativistic and relativistic energy density functionals. To assess the degree of correlation between nuclear observables and to explore systematic and statistical uncertainties on theoretical predictions, we employ the chi-square statistical covariance technique. Results: We find a strong correlation between the weak-charge form factor and the neutron radius, that allows for an accurate determination of the neutron skin of neutron-rich nuclei. We determine the optimal range of the momentum transfer q that maximizes the information content of the measured weak-charge form factor and quantify the uncertainties associated with the strange quark contribution. Moreover, we confirm the role of the electric-dipole polarizability as a strong isovector indicator. Conclusions: Accurate measurements of the weak-charge form factor of 48Ca and 208Pb will have a profound impact on many aspects of nuclear theory and hadronic measurements of neutron skins of exotic nuclei at radioactive-beam facilities.