Unmanned Aerial Vehicles (UAVs), i.e., drones, are expected to be widely used in various applications, such as parcel delivery and passenger transport, with the benefits of mitigating traffic congestion and reducing carbon emissions. In this paper, we study a UAV path planning problem under uncertain weather conditions, and design a data-driven dynamic decision support system for multiple types of UAVs. To this end, we categorize all relevant costs into three types, namely, economic, environmental, and social costs, and formulate a nonlinear two-stage stochastic programming model to establish optimal paths for UAV missions under weather uncertainty. We then discretize the nonlinear model and propose a tight linear approximation for the discretized problem to allow for a near real-time implementation. To quantify weather uncertainty, we propose a weather scenario generation algorithm to map ensemble-based weather forecast information to airspace blockage maps. With comprehensive computational studies through simulations, we show that our proposed stochastic approach can lower operating costs by an average of around 6%, where the savings increase as weather conditions become more severe and complex. We also find that, for missions operated by small UAVs, it is not sufficient to determine a path solely based on economic cost minimization, but it should rather be through total cost minimization, which involves environmental and social costs. Considering only the economic cost in the optimization may lead to much higher non-economic costs. However, for missions operated by large UAVs, it is sufficient to determine paths through economic cost optimization, as including environmental and social costs in the optimization process does not result in solutions that are much different from those obtained by considering only the economic costs. For both small and large UAVs, a path established solely through environmental or social cost minimization may not be economically sustainable, as doing so would imply very high economic costs.