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.