I created a 3D textured plane in Blender and I want to increase the thickness so that I can 3D print it. So, I just need to basically extrude it so that in stead of a rectangular plane, I will have a cube/rectangle (three dimensional) with the 3D texture on one face.
Use the displace modifier on your plane, setting the texture to your aforementioned texture. You'll need to have enough vertices on the plane for this to work - if it's still just a 4-vertex plane, either subdivide everything multiple times in Edit mode or add a higher Subdivision Surfaces modifier before the Displace modifier. Play around with the setting until your texture is elevating itself nicely out of the plane without crashing your computer (or your 3D printing company's limits).
Then in edit mode select all the vertices down (E key) and after extrusion scale all of them to end on the same height so they form a flat surface to make your box.