Roughened copper electrodes, including those derived from cuprous oxide, have long been known to exhibit an enhanced Faradaic efficiency to C2+ products during CO2 electroreduction. However, the source of this enhancement has not been rationalized mechanistically. In this work, we present a theoretical study of roughened copper electrodes derived from cuprous oxide, phosphide, nitride, and sulfide. We utilize a carefully benchmarked effective medium theory potential to develop geometric models of the roughened electrodes on an unprecedented scale. Using density functional theory with an implicit electrolyte, we determine applied bias dependent binding energy distributions for critical reaction intermediates. We apply simple thermodynamic models to evaluate the role of surface roughening on selectivity during CO2 electroreduction. We find that the manner of roughening (i.e., starting from oxide, phosphide, sulfide, or nitride) does not significantly affect the binding energy distributions found, and we suggest design rules to maximize selectivity to C2+ products on copper.