Scientific foundation models (SciFMs) are increasingly used for predictive modeling of complex systems, but their deployment is hindered by the lack of practical uncertainty quantification (UQ) and error characterization. A key challenge is that UQ approaches developed for traditional computational models do not directly transfer to large transformer-based models, where their black-box nature and potential distribution shifts pose significant hurdles. We introduce stochastic attention, a lightweight mechanism that injects controlled randomness into attention layers of transformer-based SciFMs through a single scalar hyperparameter. This induces a distribution over predictions and yields a scalable and efficient procedure for post-hoc UQ. The hyperparameter is tuned via an efficient univariate objective function to achieve well-calibrated and sharp predictive uncertainty. Our approach operates entirely at inference time, requires no retraining, and adds negligible runtime and storage overhead, making it practical for large-scale foundation models. We demonstrate our method on SciFMs for Earth system prediction, characterizing post–fine-tuning model error. We compare against probabilistic fine-tuning baselines, including non-Bayesian and approximate Bayesian methods. Results indicate that stochastic attention delivers well-calibrated, sharp uncertainty estimates and maintains competitive point forecast accuracy with minimal added complexity.