Earthquakes normally occur as frictional stick-slip instabilities, resulting in catastrophic failure and seismic rupture. Tectonic faults also fail in slow earthquakes with rupture durations of months or more, yet their origin is poorly understood. Here, we present laboratory observations of repetitive, slow stick-slip in serpentinite fault zones and mechanical evidence for their origin. We document a transition from unstable to stable frictional behavior with increasing slip velocity, providing a mechanism to limit the speed of slow earthquakes. We also document reduction of P-wave speed within the active shear zone before stick-slip evnets. If similar mechanisms operate in nature, our results suggest that higher-resolution studies of elastic properties in tectonic fault zones may aid in the search for reliable earthquake precursors.