Analysis of the melting relationships of potential core forming materials in Ganymede indicate that fluid motions, a requirement for a dynamo origin for the satellite's magnetic field, may be driven, in part, either by iron (Fe) "snow" forming below the core-mantle boundary or solid iron sulfide (FeS) floating upward from the deep core. Eutectic melting temperatures and eutectic sulfur contents in the binary Fe-FeS system decrease with increasing pressure within the interval of core pressures on Ganymede (<14 GPa). Comparison of melting temperatures to adiabatic temperature gradients in the core suggests that solid iron is thermodynamically stable at shallow levels for bulk core compositions more iron-rich than eutectic (i.e., <21 wt % S). Calculations based on high-pressure solid-liquid phase relationships in the Fe-FeS system indicate that iron snow or floatation of solid iron sulfide, depending on whether the core composition is more or less iron-rich than eutectic, is an inevitable consequence of cooling Ganymede's core. These results are robust over a wide range of plausible three-layer internal structures and thermal evolution scenarios. For precipitation regimes that include Fe-snow, we present scaling arguments that give typical Rossby and magnetic Reynolds numbers consistent with dynamo action occurring in Ganymede's core. Furthermore, by applying recently derived scaling relationships relating magnetic field strength to buoyancy flux, we obtain estimates of surface magnetic field strength comparable with observed values.