I would explain the time averaged position of an electron over time to a 15 year old as follows:
If I were somehow small enough to sit on the nucleus of a Hydrogen atom in the 1s state, then the electron’s time-averaged position in the would appear as if I was trapped inside a hollow ball of radius a_0, where a_0 is an extremely tiny number. To be precise, the diameter of the ball would be approximately 118.40 pm, which is around 422,297 times smaller than the diameter of your hair.