Convective weather, and thunderstorm development in particular, represents a major source of disruption, delays and safety hazards in the Air Traffic Management system. Thunderstorms are challenging to forecast and evolve on relatively rapid timescales; therefore, aircraft trajectory planning tools need to consider the uncertainty in the forecasted evolution of these convective phenomena. In this work, we use data from a satellite-based product, Rapidly Developing Thunderstorms, to estimate a model of the uncertain evolution of thunderstorms. We then introduce a methodology based on numerical optimal control to generate avoidance trajectories under uncertain convective weather evolution. We design a randomized procedure to initialize the optimal control problem, explore the different resulting local optima, and identify the best trajectory. Finally, we demonstrate the proposed methodology on a realistic test scenario, employing actual forecast data and an aircraft performance model.