Yes, you could 'save' the position of a moved grid by clicking the 'Origin' button in the Insert group once.
It will insert an origin handle for XYZ-coordinate system and next time you can simply hover your grid to center on this origin. In the structure tree, you can rename these origin points to something that easily tell you the location/use.
Watch the GIF for a demo. In this example, we want the grid lines to overlap with the edges of our 3D object. The left hand image is the original location the grid snaps to by default. The right hand image is after manually moving the grid. The origin handle is inserted at the centre of this relocated grid.