A fundamental problem in fault mechanics is whether slip instability associated with earthquake nucleation depends on absolute fault strength. We present laboratory experimental evidence for a systematic relationship between frictional strength and friction rate dependence, one of the key parameters controlling stability, for a wide range of constituent minerals relevant to natural faults. All of the frictionally weak gouges (coefficient of sliding friction, μ < 0.5) are composed of phyllosilicate minerals and exhibit increased friction with slip velocity, known as velocity-strengthening behavior, which suppresses frictional instability. In contrast, fault gouges with higher frictional strength exhibit both velocity-weakening and velocity-strengthening frictional behavior. These materials are dominantly quartzofeldspathic in composition, but in some cases include certain phyllosilicate-rich gouges with high friction coefficients. We also find that frictional velocity dependence evolves systematically with shear strain, such that a critical shear strain is required to allow slip instability. As applied to tectonic faults, our results suggest that seismic behavior and the mode of fault slip may evolve predictably as a function of accumulated offset.