There is nothing about pressure changes that will affect your kidneys or the stones you may have formed in them. (Kidney stones form over significant time -- you can't make a sizeable one over a weekend.)
However, the one thing about diving that affects the kidneys is fluid fluxes. People get dehydrated when they are diving in warm climates, and also experiencing immersion diuresis (the thing that makes you have to pee while you are diving). Immersion diuresis makes you make lots of urine, which will tend to wash out any stones that have been lurking in the kidneys.
If you are a stone former, you will tend to form stones. If you make calcium oxalate stones (the most common kind), then the things you can do to avoid making them is first, to stay WELL hydrated so the urine is always dilute. Second, avoid excessive intake of calcium -- calcium supplements are contraindicated in stone formers, as is excessive consumption of dairy products. Third, you can avoid rich sources of oxalate, which include large amounts of black tea and heavy consumption of dark green, leafy vegetables like spinach and kale.
Stones form over time and lurk in the kidneys. We don't really know what knocks them loose, or why people pass them at the times they do, but big fluid fluxes as a causative influence make sense. So, on a dive trip, it makes good sense to stay constantly well hydrated, so you don't experience a sudden surge in urine production.
Nothing will absolutely prevent knocking a stone loose, though. If you've been through this before, it might make sense to ask your doctor for some pain meds before you go. If your pain is absolutely typical for your stones, it's not unreasonable to control the pain with medication and strain your urine for a couple of days. If the stone doesn't pass, you will need medical attention to ensure your kidney isn't obstructed and getting damaged. But if the pain passes and you see a stone in your strainer, you're probably okay. People who have passed multiple stones often deal with things this way, and only come to the doctor if the stone doesn't pass promptly.
