It doesn't have a more standard name than "$M_{1,n}/S_n$".

Belorousski's PhD thesis contains very explicit rational parametrizations of $M_{1,n}$ for $n<11$. I would expect (but haven't checked) that a modification of his constructions would work to show rationality of $M_{1,n}/S_n$, too.

The irrationality of $M_{1,11}$ can be seen from the nonvanishing Hodge number $H^0(\overline M_{1,11},\Omega^{11})$, which comes from modular forms; it is part of the two-dimensional "motive" associated to the Ramanujan cusp form $\Delta$. This motive transforms according to the sign representation of $S_n$ and in particular it dies in the cohomology of $M_{1,n}/S_n$. So I do not see an immediate cohomological obstruction for rationality. The same is true more generally that in the middle cohomology of $\overline M_{1,n}$ one finds the motive of cusp forms for $\mathrm{SL}(2,\mathbf Z)$ of weight $n+1$ and it transforms according to the sign representation. In fact according to a paper of Consani and Faber this is the only part of the rational cohomology of $\overline M_{1,n}$ transforming according to the sign representation of $S_n$.

The Kodaira dimension of $M_{1,n}$ is $0$ when $n = 11$ and $1$ for $n \geq 12$ by a paper of Bini-Fontanari. So this is at least an upper bound for the Kodaira dimension of $M_{1,n}/S_n$.

I'm not sure which (if any) parts of the above change in positive characteristic, but you could look at a paper of Will Sawin on the non-uniruledness of $\overline M_{1,n}$ in characteristic $p$.