The correct answer is $\boxed{\frac{1}{3}}$.
The coefficient of restitution is a measure of how much energy is lost in an elastic collision. It is defined as the ratio of the relative velocity of separation to the relative velocity of approach. In this case, the relative velocity of approach is $4 – 1 = 3$ m/s, and the relative velocity of separation is $0 – v_2 = -v_2$. Therefore, the coefficient of restitution is $e = \frac{-v_2}{3}$.
In order for the 3 kg ball to come to rest, the 6 kg ball must have a velocity of at least 3 m/s after the collision. This means that the coefficient of restitution must be at least $\frac{1}{3}$.
Option A: $\frac{2}{3}$. This is not possible, because the 6 kg ball would have to have a velocity of less than 1 m/s after the collision, which is impossible.
Option B: $\frac{1}{5}$. This is not possible, because the 6 kg ball would have to have a velocity of less than 0.6 m/s after the collision, which is also impossible.
Option C: $\frac{3}{5}$. This is not possible, because the 6 kg ball would have to have a velocity of less than 0.4 m/s after the collision, which is still impossible.
Option D: $\frac{1}{3}$. This is possible, because the 6 kg ball could have a velocity of 3 m/s after the collision.
Therefore, the correct answer is $\boxed{\frac{1}{3}}$.