2008 AMC 12A Problem 8

The cube with volume $1$ has side $1$ and surface area $6.$ The larger cube has surface area $12,$ so if its side is $s,$ then $6s^2 = 12,$ giving $s = \sqrt{2}.$ Its volume is $$(\sqrt{2})^3 = 2\sqrt{2}.$$ Thus, C is the correct answer.