This approach is called "bootstrapping" and we can look up articles about it.The method I have in mind shares some of the ideas you suggest for generated simulated data. Specifically, the method repeats the following a relatively large number of times.
1. Randomly select a fraction of the data points, say for example, one half.
2. Find the value of h0 which corresponds to the least mean square fit.
Each of these h0 values is a random variable, and the collection can be used to calculate a mean and standard deviation.
The question of which estimators are better or best depends on the technical definition of "best". Among the possible interpretations of "best" are: minimum variance, unbiased, maximum liklihood, and best mean square. I don't know which, if any, of those criteria are met by the proposed bootstrap estimator. We can probably find an article about it on web.I would expect this mean to be close to the previously calculated value using all of the data. I may be mistaken, but I think it likely that the original value using all the data is a better estimate of the best value to use than the second value. It is in this sense that I used the term "mean" before the distribution was generated. One question I am not certain about is the best number of iterations/trials to use for the calculation of the standard deviation. My guess is that the same number as the original set of data points is a good choice. I would also add the square of the difference between the two mean values to the square of the calculated standard deviation.