If you're looking for a meeting place or a good stop around ½ or halfway between Euless and San Antonio, you can try Temple, TX, which is about 2 hours south of Euless, or about 2½ hours northeast of San Antonio.
Another possible place to stop near the midpoint is Harker Heights.
About ¼ or a fourth of the way, you can stop at Hillsboro, TX. This is almost an hour south of Euless.
About ⅓ or a third of the way, you can stop at Waco, TX. This is about 1½ hours south of Euless.
About ⅔ or two thirds of the way, you can stop at Austin, TX. This is about 3 hours south of Euless, or about 1½ hours northeast of San Antonio.
About ¾ or three fourths of the way, you can stop at San Marcos, TX. This is about 3½ hours south of Euless, or almost an hour northeast of San Antonio.
If you drive straight through without stopping at all, the total distance is 286 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, 18 minutes.