Meta-learning for Bayesian optimization accelerates optimization by leveraging knowledge from previous tasks, but existing methods optimize for average performance and fail on challenging outlier tasks critical in practice. These limitations become particularly severe when target tasks exhibit distribution shifts or when optimization budgets are limited in real-world applications. We introduce MetaGameBO, a hierarchical game-theoretic framework that formulates meta-learning as robust optimization through CVaR-based task selection and diversity-aware sample learning. Our approach incorporates uncertainty-aware adaptation via probabilistic embeddings and Thompson sampling for robust generalization to out-of-distribution targets. We establish theoretical guarantees including convergence to game-theoretic equilibria and improved sample complexity, and demonstrate substantial improvements with 95.7% reduction in average loss and 88.6% lower tail risk compared to state-of-the-art methods on challenging tasks and distribution shifts.