If you're looking for a meeting place or a good stop around ½ or halfway between Mesquite and Dickinson, you can try South Salem, TX, which is almost 2 hours south of Mesquite, or a little over 2 hours northwest of Dickinson.
About ¼ or a fourth of the way, you can stop at Angus, TX. This is about 1 hour south of Mesquite.
About ⅓ or a third of the way, you can stop at Kirvin, TX. This is a little over 1 hour south of Mesquite.
About ⅔ or two thirds of the way, you can stop at Hawthorne, TX. This is almost 3 hours southeast of Mesquite, or a little over 1 hour north of Dickinson.
About ¾ or three fourths of the way, you can stop at The Woodlands, TX. This is about 3 hours south of Mesquite, or about 1 hour northwest of Dickinson.
If you drive straight through without stopping at all, the total distance is 273 miles. The distances shown above don't include local roads to get into each town, so if you're planning a long road trip, make sure you factor in traffic and road conditions, rest stop breaks, and interesting detours along the way. If you're in a rush, you can do the trip with zero stops, and your total travel time would be around 4 hours, 3 minutes.