Take a <A HREF="https://en.wikipedia.org/wiki/Cauchy_distribution">Cauchy distribution</A> for $Y$: define $Y=Z/X$ with $Z$ a zero-mean Gaussian independent of $X$; then $\eta=YX=Z$ has the desired property of having a Gaussian distribution and being uncorrelated with $X$. 

The answer becomes more complicated if $X$ has nonzero mean, as explained <A HREF="https://en.wikipedia.org/wiki/Ratio_distribution#Gaussian_ratio_distribution">here.</A>