The correct answer is $\boxed{\text{B. 10 N}}$.
The centripetal force on the stone is given by $F_c = \frac{mv^2}{r}$, where $m$ is the mass of the stone, $v$ is the velocity of the stone, and $r$ is the radius of the circle. The velocity of the stone is given by $v = \omega r$, where $\omega$ is the angular speed of the stone. The angular speed is given by $\omega = 2\pi f$, where $f$ is the frequency of the circular motion. The frequency is given by $f = \frac{1}{T}$, where $T$ is the period of the circular motion.
The period of the circular motion is given by $T = \frac{2\pi}{v}$. Substituting this into the equation for the centripetal force, we get $F_c = \frac{m(2\pi f)^2}{r}$. Substituting in the values for $m$, $f$, and $r$, we get $F_c = \frac{(1 \text{ kg})(2\pi (5 \text{ rad/s})^2}{(1 \text{ m})} = 10 \text{ N}$.
Option A is incorrect because it is the centripetal force on the stone if the stone were moving at a constant speed. However, the stone is moving in a circular path, so the centripetal force is constantly changing.
Option C is incorrect because it is the centripetal force on the stone if the string were not attached to anything. However, the string is attached to something, so the centripetal force is not equal to the weight of the stone.
Option D is incorrect because it is the centripetal force on the stone if the string were twice as long. However, the string is only 1 m long, so the centripetal force is not equal to 25 N.