A whinny noise suggests to me that it's likely something in the movement of the motor or attached assembly. Careful application of WD40 might be of some use if you can get it in past the blades, and closer to the motor itself. If that worked, but only for a short time, I would probably look for a way to open it up a bit (careful to turn off power first, and not just at the wall switch) and clean and oil more deeply.
It's possible it's just dirty, but I would guess that it's a bit more likely some parts are starting to wear a bit. A slightly heavier oil (like 3-in-1 or similar), might give you slightly longer relief than WD40 but it will be harder to get into the right places.