Minimal cumulative set sum
This problem is actually related to a scheduling problem knows as "Precedence constrained scheduling to minimize weighted completion time". The problem is as follows : Given a set of jobs, where each job has a processing time (p) and weight (w) and a precedence graph is defined on the jobs. Goal is to schedule the jobs in a single machine (non-preemptive) such that the precedence constraints are statisfied and the sum of weighted completion time is minimized. The problem is NP-hard and a 2-approximation is known. 

Reduction from minimum cumulative sum problem to Precedence constrained scheduling problem :  For each element create a job with p = 1, w = 0. Also for each set create a job with p = 0, w = 1. Create the precedence graph, such that if element $e \in S$, then $e$ must be scheduled before $S$. I think this special case of the scheduling problem is also NP-hard.

See the following links,

1) http://www.win.tue.nl/~gwoegi/papers/precsum.pdf

2) http://web.engr.illinois.edu/~chekuri/papers/dam_sched.ps