Let the dimensions of the cuboid be $a$, $b$, and $c$. According to the problem, we have $a^2+b^2+c^2=4R^2$. The surface area of the cuboid is $2ab+2ac+2bc \leq 2a^2+2b^2+2c^2=8R^2$; this means the maximum value is achieved when $a=b=c$, which is when the cuboid is a cube.

Therefore, the correct answer is $\boxed{\text{A}}$.