This must have been known to the ancients, but I am having some trouble finding the references: what can be said (especially geometrically) about the normal closure of an element in a surface group? Especially an elliptic element (in an orbifold group)...
This doesn't say much about the elliptic case, but there is Hempel's theorem that the normal closure of an element $\alpha$ contains a (power of a) simple curve $\beta$ if and only $\alpha$ is a (power of a) simple curve. Furthermore, any such $\beta$ is either a power of $\alpha$ or a commutator of $\alpha$ with a simple loop $\gamma$ intersecting $\alpha$ in one point.
You should be able to glean some structural information about the normal closure from his paper.