Bike Sharing Systems (BSSs) offer a sustainable and efficient urban transportation solution, bringing flexible and eco-friendly alternatives to city logistics. During their operation, BSSs may suffer from unbalanced bike distribution among stations, requiring rebalancing operations throughout the system. The inherent uncertain demand at the stations further complicates these rebalancing operations, even when performed during downtime. This paper addresses this challenge by introducing the Robust Bike Sharing Rebalancing Problem (RBRP), which relies on robust optimization techniques to improve rebalancing operations in BSSs. Very few studies have considered uncertainty in this context, despite it being a common characteristic with a significant impact on the performance of the system. We present two new formulations and a tailored branch-and-cut algorithm for the RBRP. The first formulation is compact and based on the linearization of recursive equations, while the second is based on robust rounded capacity inequalities and feasibility cuts. Computational results based on benchmark instances indicate the effectiveness of our approaches and highlight the benefits of using robust solutions to support decision-making in BSSs.