That rather depends on the particulars, for example, do you know (or have good reasons to assume) the characteristics of your noise?
Basically you have a noisy sample and want some kind of an efficient estimator, right?
Not really. In this particular case, I'm minimizing how long it takes a simulation reach one state, so the distribution ends up looking lognormal- or Poisson-ish.
Edit: Seeing your added question, I don't need an efficient estimator in the usual sense per se. This is more about how to search the parameter space in a reasonable way to find where the minimum is, despite the noise.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.