Suppose $U$ is a normal ultrafilter on $\kappa$ of Mitchell order zero, and let $j\_U : V \to M$ be the associated embedding. Does there exist a nonstationary $X \subseteq \kappa^+$ such that $X \in M$ and $M \models X$ is stationary?


Note that if $W$ is a normal measure derived from an embedding $i : V \to N$ where $\mathcal P(\kappa^+) \subseteq N$, then $W$ gives a stationary-correct ultrapower, hence the restriction to low Mitchell order.



Stationary correctness of ultrapowers by low order measures
It is consistent that $\kappa$ has a unique normal measure (so, in particular, Mitchell minimal) and that measure's ultrapower $M$ satisfies $\mathcal{P}(\kappa^+)\subseteq M$, so, by your note, the ultrapower is stationary-correct.


The model is just the Friedman--Magidor model for a $\kappa$ with a unique normal measure with large $2^\kappa$. I don't know that this had been observed before, but it appears as Theorem 32 in my paper with Radek Honzík, [Capturing sets of ordinals by normal ultrapowers](https://arxiv.org/abs/1902.10638).


