The definition of the min_child_weight parameter in xgboost is given as the:
minimum sum of instance weight (hessian) needed in a child. If the tree partition step results in a leaf node with the sum of instance weight less than min_child_weight, then the building process will give up further partitioning. In linear regression mode, this simply corresponds to minimum number of instances needed to be in each node. The larger, the more conservative the algorithm will be.
I have read quite a few things on xgboost including the original paper (see formula 8 and the one just after equation 9), this question and most things to do with xgboost that appear on the first few pages of a google search. ;)
Basically I'm still not happy as to why we are imposing a constraint on the sum of the hessian? My only thought at the minute from the original paper is that it relates to the weighted quantile sketch section (and the reformulation as of equation 3 weighted squared loss) which has $h_i$ as the 'weight' of each instance.
A further question relates to why it is simply the number of instances in linear regression mode? I guess this is related to the second derivative of the sum of squares equation?