Let’s draw a simple sketch, like Figure 2.5, that shows you, your dog, and your friend. In our sketch, the dog runs to the right as it runs toward your friend. So the displacement of the dog points to the right, regardless of the coordinate system we choose. Because the average velocity vector points in the same direction as the displacement, the average velocity must also point to the right. This we can say with certainty, regardless of the coordinate system we choose. The sign of the average velocity, however, depends on the coordinate system we choose along the line joining you and your friend. Below the sketch, we have drawn two possible coordinate systems that we could choose for this problem. If we choose to the right to be the positive x direction, then the sign of the average velocity will be positive. If we let the positive x direction be to the left, then the sign of the average velocity will be negative. Since both you and your friend can pick the positive direction to point to either the left or the right, both of you can obtain an average velocity that is either positive or negative. Thus, the correct answer is D.
Notice that the sign of the average velocity does not depend on where you place the origin, but only on the direction you define as positive.