These trees tend to grow where other things cannot and they also tend to grow in more arid locations. I am not certain as to your exact conditions, so I will advise based on the typical environment of the mesquite tree.

First, be aware that growing grass under any tree is difficult and growing grass under a tree in conditions that mesquite normally grow it will be very difficult. You may want to look at other groundcovers or alternative solutions.

But, if you are still wanting to try, zoysia or bermuda grass would probably be your best choices. They can take the dry conditions (though you will still need to water regularly) and are often found growing in conditions that are less than ideal for grass.