The correct answer is $\boxed{\text{B}}$.
The force between two charges is inversely proportional to the square of the distance between them. This means that if the distance between the charges is doubled, the force will be $\frac{1}{4}$ of its original value. In this case, the force will be $\frac{1}{4} \times 120 N = 30 N$.
Option A is incorrect because the force is not halved when the distance is doubled. Option C is incorrect because the force is not doubled when the distance is doubled. Option D is incorrect because the force is not four times as great when the distance is doubled.