Assume $GCH$ and let $\kappa$ be $(\kappa+2)-$strong. Force with extender based Prikry forcing with interleaved collapses to make $\kappa=\aleph_\omega$ and $2^{\aleph_\omega}=\aleph_{\omega+2}.$ Call the resulting extension $V[H].$ Also let $V[G]$ be an intermediate submodel, which just adds the Prikry sequence related to the normal measure and adds collapses, so that the following holds:

(1) $V[G] \subset V[H]$ have the same cardinals and bounded subsets of $\kappa=\aleph_\omega,$

(2) $V[G] \models GCH.$

Note that there are many new $\omega$-sequences through $\kappa=\aleph_\omega$ in $V[H]\setminus V[G].$

A much stronger version of the following claim will appear in my paper "$HOD, V$ and the $GCH$" (where extender based Prikry forcing is replaced by extender based Radin forcing):

Claim $V[H]$ is a generic extension of $V[G]$ by a cone homogeneous forcing notion.

Now force over $V[H]$ to code any bounded subset of $\aleph_\omega$ into $HOD$ using a homogeneous forcing, call the extension $V[H][K].$

Now $V[G] \subset V[H][K],$ and any new $\omega-$sequence cofinal in $\kappa$ which is in $V[H]\setminus V[G]$ witnesses $\min\{\alpha: P(\alpha) \nsubseteq HOD\}=\aleph_\omega$.

Remark. In fact it seems we need much weaker assumption. All we need is to be able to add a new cofinal $\omega$-sequence in $\kappa$ which is not in $V[G],$ and it seems to me that for this just a measurable is sufficient.

Assume $GCH$ and let $\kappa$ be $(\kappa+2)-$strong. Force with extender based Prikry forcing $P$ with interleaved collapses to make $\kappa=\aleph_\omega$ and $2^{\aleph_\omega}=\aleph_{\omega+2}.$ Call the resulting extension $V[H].$ Also let $V[G]$ be an intermediate submodel, which just adds the Prikry sequence related to the normal measure and adds collapses, so that the following holds:

(1) $V[G] \subset V[H]$ have the same cardinals and bounded subsets of $\kappa=\aleph_\omega,$

(2) $V[G] \models GCH.$

Note that there are many new $\omega$-sequences through $\kappa=\aleph_\omega$ in $V[H]\setminus V[G].$

A much stronger version of the following lemma will appear in my paper "$HOD, V$ and the $GCH$" (where extender based Prikry forcing is replaced by extender based Radin forcing):

Homogeneity lemma. Assume $p,q\in P$ are such that $\pi(p)$ is compatible with $\pi(q),$ where $\pi$ is the projection map. Then there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: P/p' \simeq P/q'.$

It follows from the above lemma that $HOD^{V[H]} \subseteq V[G].$

Now force over $V[H]$ to code any bounded subset of $\aleph_\omega$ into $HOD$ using a homogeneous forcing, call the extension $V[H][K].$

Now $V[G] \subset V[H][K],$ and any new $\omega-$sequence cofinal in $\kappa$ which is in $V[H]\setminus V[G]$ witnesses $\min\{\alpha: P(\alpha) \nsubseteq HOD\}=\aleph_\omega$.

Remark. In fact it seems we need much weaker assumption. All we need is to be able to add a new cofinal $\omega$-sequence in $\kappa$ which is not in $V[G],$ and it seems to me that for this just a measurable is sufficient.