If you're looking for a meeting place or a good stop around ½ or halfway between Katy and Phoenix, you can try Saragosa, TX, which is about 7½ hours west of Katy, or about 9 hours east of Phoenix.
About ¼ or a fourth of the way, you can stop at Mountain Home, TX. This is about 3½ hours west of Katy.
About ⅓ or a third of the way, you can stop at Ozona, TX. This is a little over 5 hours west of Katy.
About ⅔ or two thirds of the way, you can stop at Las Cruces, NM. This is almost 11 hours west of Katy, or almost 6 hours east of Phoenix.
About ¾ or three fourths of the way, you can stop at Wilna, NM. This is about 12 hours west of Katy, or about 4½ hours east of Phoenix.
If you drive straight through without stopping at all, the total distance is 1,148 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 16 hours, 27 minutes.