Ask Mr. Smarty Plants

The very best grass for a totally sunny situation would be Habiturf, developed by a team led by the Lady Bird Johnson Wildflower Center. See also this How-To Article on Native Lawns: Habiturf - The Ecological Lawn.  In both cases, you will quickly see that preparation for the Habiturf planting and maintenance of it do not lend themselves to the pitter-patter of preschool feet. Much as we hate to concede this is a situation that cannot be helped by native plants, which is what the Wildflower Center and Mr. Smarty Plants are dedicated to, you may have to consider either a non-native (and potentially invasive) bermudagrass or something like pea gravel or mulch. Neither of those choices would be ideal for your purpose, as there could be a lot of skinned knees and scuffled mulch.

So, reluctantly, we refer you to Aggie Horticulture Bermudagrass. Texas A & M has long been known for developing grasses for playing fields, golf courses and generally heavy-use plants. Ordinarily, the Lady Bird Johnson Wildflower Center would not recommend anything non-native, but this is a problem with putting in a new turf of any kind, whether or not it will take extensive traffic.