Ice chest lids are notoriously poorly insulated and lose their cool. If you can't add insulation, you can make a "blanket" out of the silver bubble wrap that people use to insulate their airplane and RV windows. Wal-Mart sells small ones for your vehicle windshield, or a good hardware store often sells it off a roll by the foot. You can cut a blanket the size of the interior of your ice chest and lay it over your ice and cold contents. Two thicknesses taped together will be stiffer and double the insulation quality. It's a pain to have to move it every time you have to get in the chest, but it will help preserve your ice, it's cheap, and you don't have to modify your boat at all. The other thing you can do is keep the sunshine off the chest some way, from a wet towel to an umbrella.
Not having ice sucks! I sailed from Galveston to the Caribbean and back one year without ice. Not fun.