The use of metamodels in simulation-based robust design introduces a new source of uncertainty that we term model interpolation uncertainty. Most existing approaches for treating interpolation uncertainty in computer experiments have been developed for deterministic optimization and are not applicable to design under uncertainty. With the randomness present in noise and/or design variables that propagates through the metamodel, the effects of model interpolation uncertainty are not nearly as transparent as in deterministic optimization. In this work, a methodology is developed within a Bayesian framework for quantifying the impact of interpolation uncertainty on robust design objective. By viewing the true response surface as a realization of a random process, as is common in kriging and other Bayesian analyses of computer experiments, we derive a closed-form analytical expression for a Bayesian prediction interval on the robust design objective function. This provides a simple, intuitively appealing tool for distinguishing the best design alternative and conducting more efficient computer experiments. Even though our proposed methodology is illustrated with a simple container design and an automotive engine piston design example here, the developed analytical approach is the most useful when applied to high-dimensional complex design problems in a similar manner.