I had a kidney stone two months after starting LCHF and greatly increasing physical exercise. The consensus of the health-care professionals is that it may have been caused by exercising without enough hydration (i.e. nothing to do with the diet). Since then, I have made sure to drink at least 1 liter of water during exercise, and another half-liter during the rest of the day. I have also added a glass of water with a half-lemon squeezed into it, to my lunch (there is apparently some evidence that lemon juice can help).
Apparently there is also a random element to kidney stones, so it's hard to say anything definite.