Recent advancements in Connected Vehicle (CV) technology have prompted research on leveraging CV data for more effective traffic management. However, existing studies on CV-based signal control share a common shortcoming in that they all ignore traffic flow estimation errors in their modeling process, which is inevitable due to the sampling observation nature of CVs. This study proposes a CV data-driven robust optimization framework for traffic signal timing, accounting for both traffic flow variability and estimation errors. First, we propose a general CV data-driven deterministic optimization model (CV-DO) that can be widely applied to various scenarios, including under-/over-saturated and fixed-/real-time signalized intersections. Then, we propose a novel CV data-driven uncertainty set of arrival rates, circumventing the error-prone estimation process and accounting for both traffic flow variability errors. Finally, a CV data-driven robust optimization model (CV-RO) is formulated to explicitly handle arrival rate uncertainties. Employing the robust counterpart approach, this robust optimization problem can be converted to deterministic mixed-integer linear programming problems that can be solved efficiently with exact solutions. The evaluation results at a real-world intersection highlight the superior performance of the CV-RO model compared to the deterministic model and traditional methods across various scenarios. At different levels of traffic flow fluctuations, CV-RO can reduce delays by 5-26% compared to CV-DO at fixed-time signalized intersections with 0.1 CV penetration rate. The results on a real-time signalized network show that CV-RO can reduce 5% delays compared to the CV-DO model and 35.5% delays compared to actuated control at a 0.3 penetration rate.