jmc

Call the coordinates of point $B$ $(x,y)$. Because the coordinates of a midpoint of a line segment are the average of the coordinates of the two endpoints, we have that $\frac{3+x}{2}=2$ and $\frac{1+y}{2}=5$. Solving for $x$ and $y$ yields $x=1$ and $y=9$. Thus, point $B$ has coordinates $(1,9)$, so the product of its coordinates is $\boxed{9}$.