An analysis of the hardness of resolution of random 3-SAT instances using the Davis-Putnam-Loveland-Logemann (DPLL) algorithm slightly below threshold is presented. While finding a solution for such instances demands exponential effort with high probability, we show that an exponentially small fraction of resolutions require a computation scaling linearly in the size of the instance only. We compute analytically this exponentially small probability of easy resolutions from a large deviation analysis of DPLL with the Generalized Unit Clause search heuristic, and show that the corresponding exponent is smaller (in absolute value) than the growth exponent of the typical resolution time. Our study therefore gives some quantitative basis to heuristic restart solving procedures, and suggests a natural cut-off cost (the size of the instance) for the restart.