Great question! Water absolutely is used as a lubricant in some power generating systems. For example many Francis style turbines use their feed water as the primary lubricant and coolant of the bottom main bearing. (In the old Loeffel turbines, that bearing was typically made of lignum vitae wood, and could last up to 100 years in near constant use). Good engineering takes advantage of the cheap availability of water whenever applicable.
But water freezes and expands, blowing apart your engines, so it's not suitable for every application. This is enough of an everyday issue that automobiles are designed with "freeze plugs" and block heater ports and typically use toxic and environmentally unfriendly antifreeze chemicals in their liquid coolant systems. If your generator fails in cold weather, you don't want it to freeze solid before you can manage to repair it.
(A freeze plug, also called a Welch plug, is a core hole plug purposely made weak enough that it will fail before the block cracks due to frozen coolant. Core holes are left by the block casting process.)
And although absolutely pure water is an electrical insulator in the laboratory, in real life it's not, because even the slightest amount of contamination makes water an excellent electrical conductor. Oils, on the other hand, typically have consistently low conductivity until they are so loaded with metal contaminants that they have lost lubricity. If your generator has an oil leak, the oil won't provide a conductive path to repair technicians and other system parts.
And finally, as others have already noted, water is very corrosive, especially seawater. Since rust expands to five times the size of the steel it came from, you can't use any of the cheap iron alloys in water, which drives up the cost of materials pretty quickly.
Petroleum oils are a superb lubricant, able to be purified and recycled many times. At least as early as 1915 intelligent people were writing that burning fuels like petroleum is a criminal waste of a fantastically valuable and limited resource; it's robbing future generations in order to turn our shared treasure into atmospheric poisons. It's nice to see people like yourself thinking about these things!