If you're looking for a meeting place or a good stop around ½ or halfway between Fayetteville and Knoxville, you can try Hickory, NC, which is about 3 hours west of Fayetteville, or about 3 hours east of Knoxville.
Other possible places to stop near the midpoint include Newton, and Morganton.
About ¼ or a fourth of the way, you can stop at Greensboro, NC. This is a little over 1 hour northwest of Fayetteville.
About ⅓ or a third of the way, you can stop at Clemmons, NC. This is about 2 hours northwest of Fayetteville.
About ⅔ or two thirds of the way, you can stop at Asheville, NC. This is about 4 hours west of Fayetteville, or about 2 hours east of Knoxville.
About ¾ or three fourths of the way, you can stop at Waynesville, NC. This is about 4½ hours west of Fayetteville, or about 1½ hours southeast of Knoxville.
If you drive straight through without stopping at all, the total distance is 377 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 5 hours, 58 minutes.