Synthesizing diverse, uncertainty-aware grasps for multi-fingered hands from partial observations remains a critical challenge in robot learning. Prior generative methods struggle to model the intricate grasp distribution of dexterous hands and often fail to reason about shape uncertainty inherent in partial point clouds, leading to unreliable or overly conservative grasps. We propose FFHFlow, a flow-based variational framework that generates diverse, robust multi-finger grasps while explicitly quantifying perceptual uncertainty in the partial point clouds. Our approach leverages a normalizing flow-based deep latent variable model to learn a hierarchical grasp manifold, overcoming the mode collapse and rigid prior limitations of conditional Variational Autoencoders (cVAEs). By exploiting the invertibility and exact likelihoods of flows, FFHFlow introspects shape uncertainty in partial observations and identifies novel object structures, enabling risk-aware grasp synthesis. To further enhance reliability, we integrate a discriminative grasp evaluator with the flow likelihoods, formulating an uncertainty-aware ranking strategy that prioritizes grasps robust to shape ambiguity. Extensive experiments in simulation and real-world setups demonstrate that FFHFlow outperforms state-of-the-art baselines (including diffusion models) in grasp diversity and success rate, while achieving run-time efficient sampling. We also showcase its practical value in cluttered and confined environments, where diversity-driven sampling excels by mitigating collisions (Project Page: https://sites.google.com/view/ffhflow/home/).