I keep the lawn for a few reasons. First and foremost, my one-year old daughter absolutely loves being outside and always have. She was out crawling in the yard at six or eight months old. It's allowed her to get a nice amount of sun and outside play without worrying about her falling on rocks and gravel.
I also like the grass because it helps insulate the ground against the sun's heat. During these extreme temps, dirt and gravel can both get extremely hot. A plot of grass is comfortable to walk on no matter what the temperature. All of my girls are bedding down on the grass in the evenings and being lazy. I've never found them that way in the dirt areas, nor the rocks, even though the rock landscaping is all under the shade of the oleander. They are more often sitting in the sunny grass versus the shaded rock. Going hand-in-hand with that, I think it decreases the residual heat that is let off at night, too, allowing the backyard to remain cooler at all times. We have way to much of a "parking lot effect" as it is. If I can reduce the temperature in my immediate area without any additional effort or very little expense, I'm game. To be quite honesty, I spend more time and effort trying to keep my darn gravel front yard clean. All the leaves, trash and everything imaginable gets caught in the rocks. A rake didn't work well at all. For a leaf vacuum or blower, you are either using larger amounts of electricity or gasoline.
Of course, you have the extra benefits of grass for everyone. Fresh greens and more bugs for the chickens, the amazingly beautiful aesthetic looks, the extra material for composting if necessary, gentleness on the body for the kids playing, etc. A desert landscape can't accomplish any of this.