jmc

Consider the point $(x,y)$. Then, he labels the point with the number $$(\sqrt{(x-0{)}^2+(y-0{)}^2}{)}^2=x^2+y^2,$$ so it follows that $x^2+y^2=25$. From here, some casework needs to be done to find the number of pairs $(x,y)$ that satisfy $x^2+y^2=25$. We note that $x^2=25-y^2\le 25⟹|x|\le 5$, and so $|x|$ can only be equal to $0,1,2,3,4,5$. Of these, only $0,3,4,5$ produce integer solutions for $|y|$.

If $|x|=3$, then $|y|=4$, and any of the four combinations $(3,4)(-3,4)(3,-4)(-3,-4)$ work. Similarly, if $|x|=4,|y|=3$, there are four feasible distinct combinations.

If $|x|=0$, then $|y|=5$, but then there is only one possible value for $x$, and so there are only two combinations that work: $(0,5)$ and $(0,-5)$. Similarly, if $|x|=5,|y|=0$, there are two feasible distinct combinations. In total, there are $\boxed{12}$ possible pairs of integer coordinates that are labeled with $25$.