Hand Tools Archive
Is it oil based or water based? It probably doesn't really matter.
Any thin lubricant will work well on the fine stones, as will anything up to medium mineral oil. On the heavier stones (like medium crystolon or more coarse, especially if friable, an oil bath or thicker oil like light to medium mineral oil at least - is needed to keep the stone from loading).
What you use or don't use is more about feel - what you like or don't like. Some people love dish soap-based water on oilstones. I hate it. Some people don't like WD-40 on medium and fine stones (like fine india or finer) - I really like it.