A multiset $S$ of numbers is considered satisfactory if $\left|\left\{\sum U \bmod n \mid U\subseteq S\right\}\right| = n$, that is, all remainders mod $n$ can be constructed by summing subsets of $S$.
If we start a random process with an empty multiset, and keep adding uniformly random integers $[0, n)$ (with replacement) until it is satisfactory, how many numbers will we have added on average?
Or conversely, if that question is easier, if we have a collection of $k$ random numbers, what is the probability that it is satisfactory?
Heuristically, I suspect something like $\ell \cdot H_\ell + O(\ell)$ numbers should suffice, where $\ell = \lg |S|$ and $H_\ell$ is the $\ell$th harmonic number. In particular, there are $2^{|S|}$ subsets of $S$, and if we model each as having a random sum mod $n$, then the coupon collector problem gives us this estimate for the number of draws until we have obtained all possible remainders.
Of course this heuristic makes an independence assumption that is totally bogus, so this is not by any means a rigorous result. It's possible this estimate might be terribly wrong if there is some interesting structure that throws things off. But I would speculate it is probably in the right ballpark.