The cleanest notation that works for all metric signatures in 4D is found in
Wikipedia:
$$\tag{1}
\star\,(dx^\mu)=\eta^{\mu\lambda}\varepsilon_{\lambda\nu\rho\sigma}\frac1{3!}dx^\nu\wedge dx^\rho \wedge dx^\sigma\,.
$$
I just learned here that there might be different conventions for the sign of the Levi-Civita symbol but thankfully Wikipedia mentions clearly that in (1) they use the convention $\varepsilon_{0123}=1\,.$
- Now for the metric $\eta^{\mu\lambda}$ assume West coast convention $(+---)$ which is popular in particle physics.
By the anti symmetry of $\varepsilon_{\lambda\nu\rho\sigma}$ and the anti symmetry of
$dx^\nu\wedge dx^\rho \wedge dx^\sigma$ in all indices it is clear that
when summing over $\nu,\rho,\sigma$ we can just take one term
where $\nu<\rho<\sigma$ and drop the factor $\frac1{3!}\,.$
That is because any permutation in an index pair leads to a sign change of the form $(-1)(-1)=1\,.$ Another drastic simplification arises because $\eta^{\mu\lambda}$ is diagonal.
For $\mu=2\,,$ we have $dx^\mu=dy$ and therefore
\begin{align}\tag{2}
\star\,dy&=
\underbrace{\eta^{22}\varepsilon_{2013}}_{(-1)(+1)}\, dt \wedge dx\wedge dz=-\,dt\wedge dx\wedge dz
\end{align}
which is the answer that you said is correct.
It is quite obvious, too, that with the East coast convention $(-+++)$ and Levi-Civita symbol normalized to $\varepsilon_{0123}=1$ we get a plus sign:
$$\tag{3}
\star\,dy=dt\wedge dx\wedge dz
$$
which agrees with WP because they write this as
$$\tag{4}
\star\,dy=-\,dt\wedge\color{red}{dz}\wedge\color{red}{dx}\,.
$$
