If the observations are independent, and the ordering of the observations is unimportant, then you could simply estimate the mean as Actuary suggests (+1); however the fact that multiple observations are made suggests that the spread of the observations may be of intrinsic interest, in which case it would be a good idea to model the distribution of plausible values for the observation (for a Gaussian, this would be modelling the conditional variance as well as the conditional mean).
I did some work on predictive uncertainty a few years ago, and the method described in
G. C. Cawley, N. L. C. Talbot, R. J. Foxall, S. R. Dorling and D. P. Mandic, Heteroscedastic kernel ridge regression, Neurocomputing, vol. 57, pp 105-124, March 2004.
might be of some interest (MATLAB demo here). It ought to be fairly straightforward to make a basic multiple instance version assuming the observations are conditionally i.i.d.
BTW I think this kind of problem is known as "multiple instance learning" in machine learning, there is a fair amount of research on this indexed by Google Scholar, so hopefully there may be something relevant to your problem there somewhere (too many papers, too little time to read them... ;o).