In this article we study some Liouville-type theorems for the stationary 3D Navier-Stokes equations. These results are related to the uniqueness of weak solutions for this system under some additional information over the velocity field, which is usually stated in the literature in terms of Lebesgue, Morrey or BMO^−1 spaces. Here we will consider Lebesgue spaces of variable exponent which will provide us with some interesting flexibility.