Thanks for your suggestions. For my use case (tuning parameters of a financial market simulation), I'm essentially able to get good noise estimates for free by re-sampling a set of parameters multiple times.

So for example, rather than simulate an entire month in one shot, I'll simulate a day 30 times and therefore have a decent estimate of the noise for that result and be able to clearly distinguish the noise from the covariance of the Gaussian process.

The noise in these simulations can vary dramatically in parameter space (easily 10-100x), so it seems like it would be important to model.