Well, you've received some good ideas from folks but it seems we have more info now. You sell from your house but the customers can't come to your house...that makes it rather difficult to sell them.
An idea would be to sell them via social areas that you visit....church, school, ballgames, whatever gatherings of people that you or your family attend are targets areas. Get the word out via word-of-mouth, maybe some flyers to hand to people, or a simple business card. On it state that you have farm fresh eggs, if you free range then add that info, and that you will deliver to such-and-such gathering. You'll have to include a phone number to make it work well, which might be a problem with your mother, though. Otherwise, simply state on the hand-out that you will have eggs at whatever function you attend. Maybe if there is a farmers market nearby you could set up a table there or either sell through another vendor....be kind of like a supplier to the retail seller.
Also, if you were to put a small rack in the cooler to set the egg cartons on and keep the ice below the rack that would probably help keep the egg cartons from getting soggy.
Without a sign out front and people coming to your door it will be somewhat difficult to sell from your house unless you use an honor system with the ice cooler.
Best wishes with your business enterprise!!!!!!!!!!!
Ed