Dark matter lighter than 10 GeV/c$^2$ encompasses a promising range of candidates. A conceptual design for a new detector, DarkSide-LowMass, is presented, based on the DarkSide-50 detector and progress toward DarkSide-20k, optimized for a low-threshold electron-counting measurement. Sensitivity to light dark matter is explored for various potential energy thresholds and background rates. These studies show that DarkSide-LowMass can achieve sensitivity to light dark matter down to the solar neutrino floor for GeV-scale masses and significant sensitivity down to 10 MeV/c$^2$ considering the Migdal effect or interactions with electrons. Requirements for optimizing the detector's sensitivity are explored, as are potential sensitivity gains from modeling and mitigating spurious electron backgrounds that may dominate the signal at the lowest energies.