The sharpening specialists will sell you tiny amounts of their recommended oils at grossly over-inflated prices.
Mostly what is needed is something that'll keep the pores in the stones from clogging.
Kerosene works. 3-in-1 works, mineral oil works. The modern low-viscosity synthetic multigrade motor oils work.
The stuff I used to use for preference was Liberon Lib-Net, which became Liberon Wax and Polish remover, and is/was probably only readily available in the UK and/or Europe: it seemed to have just about the right amount of lubricity, didn't dry or polymerize, kept the stones nice and clean, and was available in 1l (quart) tins at reasonable cost.
I went over to water stones for most things a decade or more back and hardly ever use oilstones now. When I do, it's with either 3-in-1 or Wahl clipper oil simply because those are what tend to be handy.
I'm only slightly **** about sharpening and I'm sure there will be someone along shortly who is deeply obsessive about it. My experience has been that it doesn't matter much what you use on India oilstones or Arkansas stones, so long as it is a fairly thin oil.