High pitched humming noise from rear near side

Gear oil won't make the humming noise, lack of it will. Depending on type of vehicle you're talking about.

Possible wheel bearing getting noisy. Most automotive repair shops will have a technician test drive the vehicle free of charge. A wheel bearing noise can be easily identified by a well trained tech

maybe some thing in the brakes

That is very likely a wheel bearing staring to go out to lunch.

Does it make a noise when in 4x4 only? Does it make the noise in 2WD as well? A humming noise is usually associated with a wheel bearing noise. If while driving you steer from side to side and the noise changes pitch or becomes less/more severe it is most likely a wheel bearing. You can raise the vehicle and safely support it. Grab top and bottom of the wheel and feel for play.

Does the humming noise reduce or change in pitch slightly when turning a corner?
Most likely a wheel bearing failure. Also, when one wheel bearing on one side fails, the other is not often far behind. I often therefore replace in pairs.

are your tires cupped or dipped, have you checked rear wheel bearings ??

Check the CV joints for sloppiness and if a rubber boot is torn, allowing road grit to enter, then replace the CV joint.

ok it could be two things. it could be rear wheel bearings or a rear diff. If its a high pitch wine then its worn gears in your differential. its needs serviced or replaced. If it more of a rumbling or grinding hum, coming only from one side or both) then there are bad wheel bearings.

It needs a wheel bearing. You should notice that the pitch of the noise changes as you change the speed of the car. Also the noise is changes when you brake and/or turn, it may even disappear all together.