My favourite for using on an oilstone is water with a little dish soap (Fairy Liquid Eco-Ver or even ecconomy basic)
The rubbing action on the stone produces foam that clears the swarf and helps prevent pinning. Discovered this while polishing badly 'dragged' brass clock plates. Qiick wash and allow to dry the stone after using. If used with steel tools they need a light coat of oil immediately after. Rust forms surprisingly fast on bare steel!
To bring a stone back to flat and revitalise a blunt glazed surface I use corborundum powder mixed with water and soap solution on a glass plate. The grit rolls as the stone is rubbed over the plate and the sharp grain edges attack the stone bonding releasing the blunt stone surface grains rather than blunting the surface even more (big problem using diamond plates with fixed particles)
Its a bit like using a rotating wheel dressing stick on a bench grinder rather an old piece of stone. It tears and opens the surface rather than polishing it.
Just an opinion but to me oil shouldn't go anywhere near a wheel or stone. There are specialist soluble 'oils' used on surface or universal grinders but they are not oily, they more resemble antifreeze type fluids that include rust inhibitors.
there are a lot of razor guys who don't like oil based stuff. I don't know why. It's my opinion that the action of the stones is a little better if oil is used, but they can certainly be used without.
The other side benefit is that the use of a hydrotreated mineral oil will probably much stop tool rusting, even for a very long period of time.
I also flatten them when they're new - fine work is more easily done with flat stones, and judicious use keeps them flat just with minor periodic conditioning.
One of the favorites on the US forums to get oil like feel without using something that's not water based was ballistol. I still have a tube of it and never used it.
ballistol does have some viscosity - no clue on the name of it but would assume it's for guns or something.