Got into an argument with a family member over this. The water in the cooler was still ice cold. In fact, there was still some unmelted ice left. My Aunt insisted that it was better to drain the water from the cooler and then add ice. I explained that water has a higher specific heat capacity than air, and should be kept in, as long as the cooler had enough space. Because of water's higher specific heat capacity, it could absorb more heat from the environment, before increasing in temperature, than an air and ice mixture could do.