Consider probability distribution which is unimodal, symmetric and mean value exists. (Of course due to symmetry mean will coincide with mode (position of the maximum)).
Question Is sample mean always the "best" estimate of the mean or may be median, whatever can be better ?
Situation: assume the "tails" are heavy, (but still mean exists), so "sample-mean" will not be very stable, because big deviations will occur with big probability and hence mean will deviate of the "ideal" value. Median estimate seems to be more stable - if big deviations happens rarely median will not be affected by this.
So may be one can "cook up" something in the middle between mean and median - just filter out values which are at the ends and take average of the others or something like that...