Answer
$a^3\sqrt{1-u^2/c^2}$.
Work Step by Step
The proper length of a side is “a”. The side parallel to the direction of motion is shortened, by Lorentz contraction, to $a\sqrt{1-u^2/c^2}$. The lengths of the other 2 sides of length “a”, moving perpendicular to the motion, are unaffected.
The volume is the product of the 3 perpendicular sides, so S’ measures the volume as $ a^3\sqrt{1-u^2/c^2}$.
