A transversal for a quotient group $G/N$ is a subset of $G$ which contains precisely one element for each coset of $G/N$. This can be defined analogously for any collection of sets and not just cosets (e.g. $N$ need not be normal, so "left-transversal" and so on), but lets just talk about quotient groups because this is a terminology question and terminology often change between subject areas.
My question is: I have a subset $S$ of $G$ which contains a transversal of $G/N$. Is there any nice, snappy name (in the same vein as "transversal") I can use for this property of $S$: the set $S$ is a XXX for $G/N$?
"$S$ is a set of coset representatives" is not precise enough - it allows the possibility of cosets without a representative in $S$. "$S$ is a subset of $G$ which contains a transversal for $G/N$" or even "$S$ is a subset of $G$ which contains an element from each coset" are not snappy.
I nowhere saw or heard from such a notion.
(I choosed a wording close to what Kurzweil & Stellmacher write in their book Theory of Finite Groups, but for them a transversal is always by default a right transversal).
Now you could define a transversal to be a set which is simultanously a left and a right transversal. In a finite group for each subgroup we can find such an transversal (which is a consequence of Hall's marriage theorem), but what makes normal subgroups special here is that they have the property that every left transversal is a right transversal (finite or not). So you could safely just say transversal in the normal case, i.e. let $S$ be a transversal for $N$ in $G$. But if you like to be more specific you could call it "normal transversal" or "transversal for normal subgroup".