If a circle is centered on the origin and has radius r, it may pass through some points with integer coordinates. In fact, by symmetry, this will be a multiple of 4.
Find the smallest radius that will pass through 4n integer points where n=1,2,3,4,5.
Feel free to go further.
The smallest radius I see for n = 4 is the sqrt(65), which goes through (1,8),(8,1),(4,7), and (7,4).
Putting it all together, my answer (which might be wrong) is:
n r r^2
-- -- -----
1 1 1
2 sqrt(5) 5
3 5 25
4 sqrt(65) 65
5 25 625
I am struck by all the 5's. Probably because of 3-4-5 and 5-12-13 triangles. Note that 65 = 5*13.
n = 4 + full solution (spoiler)
