Why would you charge less? It's a highly specialized service that is nearly impossible to find. Personally, I'd capitalize on that and charge more. Don't forget, if the child is awake for whatever reason, so are you.....and you still have to function for your day job I have no idea where to advertise other than here and other websites like this. I would probably have a separate listing for it (almost like a separate business) that addresses only the overnight care. It might be a good idea to fiddle with the wording on your website so if anyone's looking for overnight care, your place comes up first in a search. Beyond that, I have no useful advice.