I have a couple year old two handle bathroom faucet that I'm having an issue with that I don't understand.

When my son is playing or washing his hands, sometimes he puts his hands over the aerator restricting the water flow. When he does this, the water is diverted and ends up coming out the hole between the two handles (under the sink) where the drain plunger actuator cable comes out.

Any ideas why this happens?

Most likely the water is just running down the spout and under the base of the fuacet. If the fuacet base seal isn't good (plumbers putty or rubber seal) the water will just leak down through the hole in the sink. Another option is the aerator gasket is shot, or the aerator isn't tight enough. Final option, the faucet has a leak internally and would need to be replaced.