Recent observations support the suggestion that short-duration gamma-ray bursts are produced by compact star mergers. The x-ray flares discovered in two short gamma-ray bursts last much longer than the previously proposed postmerger energy-release time scales. Here, we show that they can be produced by differentially rotating, millisecond pulsars after the mergers of binary neutron stars. The differential rotation leads to windup of interior poloidal magnetic fields and the resulting toroidal fields are strong enough to float up and break through the stellar surface. Magnetic reconnection-driven explosive events then occur, leading to multiple x-ray flares minutes after the original gamma-ray burst.