If you're looking for a meeting place or a good stop around ½ or halfway between Abilene and Mesa, you can try Juarez, Mexico, which is a little over 6 hours west of Abilene, or about 6½ hours east of Mesa.
About ¼ or a fourth of the way, you can stop at West Odessa, TX. This is almost 3 hours west of Abilene.
About ⅓ or a third of the way, you can stop at Hermosa, TX. This is almost 4 hours west of Abilene.
About ⅔ or two thirds of the way, you can stop at Guess Place, NM. This is a little over 8 hours west of Abilene, or about 4½ hours east of Mesa.
About ¾ or three fourths of the way, you can stop at Gary, NM. This is about 9 hours west of Abilene, or about 3½ hours east of Mesa.
If you drive straight through without stopping at all, the total distance is 884 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 12 hours, 41 minutes.