Solar contamination, due to moonlight and atmospheric scattering of sunlight, can cause systematic errors in stellar radial velocity (RV) measurements that significantly detract from the ∼10 cm s-1 sensitivity required for the detection and characterization of terrestrial exoplanets in or near habitable zones of Sun-like stars. The addition of low-level spectral contamination at variable effective velocity offsets introduces systematic noise when measuring velocities using classical mask-based or template-based cross-correlation techniques. Here we present simulations estimating the range of RV measurement error induced by uncorrected scattered sunlight contamination. We explore potential correction techniques, using both simultaneous spectrometer sky fibers and broadband imaging via coherent fiber imaging bundles, that could reliably reduce this source of error to below the photon-noise limit of typical stellar observations. We discuss the limitations of these simulations, the underlying assumptions, and mitigation mechanisms. We also present and discuss the components designed and built into the NEID (NN-EXPLORE Exoplanet Investigations with Doppler spectroscopy) precision RV instrument for the WIYN 3.5 m telescope, to serve as an ongoing resource for the community to explore and evaluate correction techniques. We emphasize that while "bright time" has been traditionally adequate for RV science, the goal of 10 cm s-1 precision on the most interesting exoplanetary systems may necessitate access to darker skies for these next-generation instruments.