"More electrons will land on one side than the other, by chance."
I think they land on the side that faces the negative pole of the current source. It's not random. They get stuck there for a while. They are repelled by the negative pole, so the drop will start turning.
"When they reach the bottom of the droplet, some will fall off and continue moving through the oil."
Actually, the droplet is also a conductor, albeit slower than the oil. So the current inside the droplet will slowly neutralize these surface charges--as the droplet turns.
The key is that there is a difference in conductivity and permeability between the two mediums.