Another way of looking at it is that the term 1/2 * rho * V^2 is the dynamic pressure. The 1/2 is there to ensure that the term is comparable with the static pressure.
Along any streamline the static pressure and the dynamic pressure sum to a constant. So where the velocity is higher, the dynamic pressure is higher and hence the static pressure is lower - Bernoulli's principle as we all know it.
If we didn't have the 1/2 term, then this relationship would not hold.
Of course, this relates to what Mathew Parsons was saying:
The 1/2 rho v squared is essentially the kinetic energy density or kinetic energy per unit volume.
Static pressure is the the kinetic energy density associated with the random thermal motion of the air molecules. Dynamic pressure is the kinetic energy density associated with the non-random motion of the air molecules.
Daniel