The LSST survey will provide unprecedented statistical power for measurements of dark energy. Consequently, controlling systematic uncertainties is becoming more important than ever. The LSST observing strategy will affect the statistical uncertainty and systematics control for many science cases; here, we focus on weak lensing systematics. The fact that the LSST observing strategy involves hundreds of visits to the same sky area provides new opportunities for systematics mitigation. We explore these opportunities by testing how different dithering strategies (pointing offsets and rotational angle of the camera in different exposures) affect additive weak lensing shear systematics on a baseline operational simulation, using the $\rho-$statistics formalism. Some dithering strategies improve systematics control at the end of the survey by a factor of up to $\sim 3-4$ better than others. We find that a random translational dithering strategy, applied with random rotational dithering at every filter change, is the most effective of those strategies tested in this work at averaging down systematics. Adopting this dithering algorithm, we explore the effect of varying the area of the survey footprint, exposure time, number of exposures in a visit, and exposure to the Galactic plane. We find that any change that increases the average number of exposures (in filters relevant to weak lensing) reduces the additive shear systematics. Some ways to achieve this increase may not be favorable for the weak lensing statistical constraining power or for other probes, and we explore the relative trade-offs between these options given constraints on the overall survey parameters.