A

Step 1: Consider a stone of mass $m$ tied to a string of length $L$, rotating in a vertical circle. The tension in the string and the gravitational force must provide the necessary centripetal force at the top of the circle.
Step 2: At the top of the circle, the forces acting on the stone are its weight ($mg$) and the tension ($T$) in the string. The centripetal force required to keep the stone moving in a circle of radius $L$ at speed $v$ is given by: $$F_c = \frac{mv^2}{L}$$
The equation for forces at the top of the vertical circle can be set up as: $$mg + T = \frac{mv^2}{L}$$
Step 3: For the minimum speed at the top, the tension in the string can be minimized (approaching zero). Hence, we let $T = 0$: $$mg = \frac{mv^2}{L}$$
This simplifies to: $$g = \frac{v^2}{L}$$
Step 4: Rearranging gives: $$v^2 = gL$$
Taking the square root: $$v = \sqrt{gL}$$
Step 5: The minimum speed $v$ is dependent only on the gravitational acceleration $g$ and the length of the string $L$, but it does not depend on the mass $m$ of the stone. Therefore, as the string's length increases, the speed also increases, but the mass of the stone does not affect this minimum speed.
Conclusion: Hence, the correct answer is A: Is independent of the mass of the stone.