We report a study of the eclipse timing variations in contact binary systems, using long-cadence lightcurves from the Kepler archive. As a first step, observed minus calculated (O - C) curves were produced for both the primary and secondary eclipses of some 2000 Kepler binaries. We find similar to 390 short-period binaries with O - C curves that exhibit (1) random walk-like variations or quasi-periodicities, with typical amplitudes of +/- 200-300 s, and (2) anticorrelations between the primary and secondary eclipse timing variations. We present a detailed analysis and results for 32 of these binaries with orbital periods in the range of 0.35 +/- 0.05 days. The anticorrelations observed in their O - C curves cannot be explained by a model involving mass transfer, which, among other things, requires implausibly high rates of similar to 0.01 M-circle dot yr(-1). We show that the anticorrelated behavior, the amplitude of the O - C delays, and the overall random walk-like behavior can be explained by the presence of a starspot that is continuously visible around the orbit and slowly changes its longitude on timescales of weeks to months. The quasi-periods of similar to 50-200 days observed in the O - C curves suggest values for k, the coefficient of the latitude dependence of the stellar differential rotation, of similar to 0.003-0.013.