It looks to me like spider mite damage, although they don't usually go after herbs. IF you run your finger along the underside of the leaf, does it feel sandpapery? In any case, there is an easy, cheap, non-toxic solution to try - soapy water spray.
Mix up 1/2 teaspoon of dish soap (not dishwasher detergent, liquid soap you use in the sink) with a quart of water in a clean spray bottle. Spray the plants thoroughly, making sure to hit the undersides of the leaves especially. Spray again at least twice a week, also after watering or it rains, as the soap will rinse off. Do this for a couple of weeks and see if the damage stops. You need to repeat the spray a few times, hence the two weeks time frame because even if you kill off the adult mites, the eggs will continue to hatch for a week or two.
Be sure to rinse off the leaves before you use them. They will be nice and clean.
The only other thing that might make spots like that is overspray of some chemical blown in from a neighbor's yard. But that wouldn't be hit or miss, every leaf would show damage and it would only happen once, not continuously. Also it wouldn't have turned up on the plant your cousin potted up.
