Mars lacks an internally generated magnetic field today. Crustal remanent magnetism and meteorites indicate that a dynamo existed after accretion but died roughly four billion years ago. Standard models rely on core/mantle heat flow dropping below the adiabatic limit for thermal convection in the core. However, rapid core cooling after the Noachian is favored instead to produce long-lived mantle plumes and magmatism at volcanic provinces such as Tharsis and Elysium. Hydrogenation of the core could resolve this apparent contradiction by impeding the dynamo while core/mantle heat flow is super-adiabatic. Here we present parameterized models for the rate at which mantle convection delivers hydrogen into the core. Our models suggest that most of the water that the mantle initially contained was effectively lost to the core. We predict that the mantle became increasingly iron-rich over time and a stratified layer awaits detection in the uppermost core—analogous to the E’ layer atop Earth’s core but likely thicker than alternative sources of stratification in the Martian core such as iron snow. Entraining buoyant, hydrogen-rich fluid downwards in the core subtracts gravitational energy from the total dissipation budget for the dynamo. The calculated fluxes of hydrogen are high enough to potentially reduce the lifetime of the dynamo by several hundred million years or longer relative to conventional model predictions. Future work should address the complicated interactions between the stratified, hydrogen-rich layer and convection in the underlying core.