Merging neutron stars produce “kilonovae”—electromagnetic transients powered by the decay of unstable nuclei synthesized via rapid neutron capture (the r-process) in material that is gravitationally unbound during inspiral and coalescence. Kilonova emission, if accurately interpreted, can be used to characterize the masses and compositions of merger-driven outflows, helping to resolve a long-standing debate about the origins of r-process material in the Universe. We explore how the uncertain properties of nuclei involved in the r-process complicate the inference of outflow properties from kilonova observations. Using r-process simulations, we show how nuclear physics uncertainties impact predictions of radioactive heating and element synthesis. For a set of models that span a large range in both predicted heating and final abundances, we carry out detailed numerical calculations of decay product thermalization and radiation transport in a kilonova ejecta with a fixed mass and density profile. The light curves associated with our models exhibit great diversity in their luminosities, with peak brightness varying by more than an order of magnitude. We also find variability in the shape of the kilonova light curves and their color, which in some cases runs counter to the expectation that increasing levels of lanthanide and/or actinide enrichment will be correlated with longer, dimmer, redder emission.