Short answer: Look for papers on "deequivariantization". (I think the original references are by Müger and Brugières, but I am not sure whether they used the term "deequivariantization".)
Longer answer:
The procedure mentioned in the paper is actually a sort of predual to the deequivariantization procedure found in the literature, in the sense that $Rep(C/S) \cong Deeq(Rep(C))$. So it is very closely related to deequivariantization, but not exactly the same thing.
You attempt was on the right track. One starts with $C$, then adds isomorphisms between objects of $S$ and the trivial object. This has to be done carefully, of course. One uses the fact that $S \cong Rep(G)$ for some finite group $G$, and $Rep(G)$ has a fiber functor.
A fancier way of thinking of it goes as follows. Since $S$ is symmetric monoidal we can think of it as an $n$ category-category for any $n$, and in particular we can thingthink of it as a 4-category. The 3-category $C$ then becomes a module 3-category for the 4-category $S$. We can construct another module 3-category $F$ for $S$ by using the isomorphism $S\cong Rep(G)$ and the fiber functor for $Rep(G)$. Now we can now define $$ C/S = C \otimes_S F $$
$F$ is actually a $S$-$G_4$ bimodule 3-category, where $G_4$ denotes the finite group $G$ thought of as a 4-category. It follows that $C/S$ has a $G$ action. (A $G$ action on a 3-category is the same thing as a $G_4$-module structure on that 3-category.)
I have a preprint which fills in the details of the above arguments, but it is not yet ready for the arXiv. If you go to this page and look at the slides from talks at Vienna and Princeton (Feb 2014), you can find some of the details.