Your son is certainly to be commended for his reasoning. But once some details about the multiverse hypothesis are considered, it's clear that such an argument doesn't apply.
First, what physicists typically mean when they talk about the multiverse is not the "parallel universe" idea common to science fiction like Star Trek or Fringe. Rather, the multiverse is rooted in inflationary cosmology. In the multiverse scenario, different parts of the universe are allowed to inflate independently, possibly at different rates. The "bubbles" that subsequently form can be casually disconnected from each other and therefore observers in separate regions can't communicate with one another. In the cosmological parlance, they are separated by cosmic horizons. An observer within one of these horizons counts the space within the horizon as his "observable universe." The actual universe, which contains all of these regions, is then called the "multiverse", while the observable universe regions are called "universes." Obviously this nomenclature can be easily confusing when the details are lacking.
The causual disconnect between universes makes it clear that the properties of a single universe can't have any effect whatsoever on the other universes. But it is also important to understand what physicists mean that was interpreted as "one universe for every possibility." The "possibilities" that are actually considered are fairly concrete. They are things like the physical constants: the gravitational constant, the fine structure constant, the masses of particles. etc. They can also involve some more complex features such as the structure of the gauge group of fundamental interactions. Basically what is allowed to vary between universes are the laws of physics and the parameters that appear in them. Existence of a multiverse is not the type of "possibility" that varies between universes. The existence of the multiverse is already the building block of the model, so it's assumed.
Other types of possibilities, such as the existence of Leonard Nimoy (with or without sinister goatee) are also not considered. The anthropic principle so often applied to the multiverse is so far only concerned with the physical laws and whether or not nuclei, atoms and stars could have formed in such a way to allow any sort of human to exist in a given universe.
It's certainly wise to be skeptical of the multiverse scenario. However, it's unlikely that it's possible to rule it out without having the correct "microscopic" description of the cosmological evolution of the universe. It's also probably good to note the difference between the multiverse concept and the anthropic principle usually applied to it. The latter might attract a larger amount of skepticism than is quite deserved by the former.
