Since the new circle has twice the area of the original circle, its radius is $\sqrt{2}$ times the old radius.
Thus,
$$r + n = r\sqrt{2}$$
$$n = r\sqrt{2} - r$$
$$n = r(\sqrt{2} - 1)$$
$$r = \frac{n}{\sqrt{2} - 1}$$
Rationalizing the denominator yields
$$r = \frac{n}{\sqrt{2} - 1} * \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = n(\sqrt{2} + 1)$$
Therefore, the answer is $\fbox{(A)}$