I don't know if there's an actual reason, but I know this : one of the basic reasons for the existence of groups in many fields of science is to study symmetries of objects. There is this theorem that justifies the use the term "symmetric" for $S_A$ :
Cayley's theorem. Let $(G,\cdot)$ be a group. Then $(G,\cdot)$ is isomorphic to a subgroup of $S_G$, where $S_G$ is the group of all bijections from $G$ to $G$ (here $G$ is considered the underlying set and $(G,\cdot)$ is the group, for clarification).
So in some way any group is a subgroup of $S_A$ for some set $A$, hence any symmetry over some object represented by a group structure lies "somewhere in $S_A$ for some $A$. That is why I agree with the name "symmetric group" and I think it should stay that way, but I have no idea why historically we have been naming it like that.
Hope that helps,