I'd try a mineral oil on a test section. Stone is porous and will absorb so it should make the white area look darker. Mineral oil is often used for soap stone counter tops.
I'd go with something like the following link. Then you'll have more if you need to do this again. You might end up coating the entire surface of the blue stone with it. It doesn't go bad and this is food safe. Baby oil is mineral oil too so the larger grocery store chains should have it as well if you wanted a smaller bottle.