Round Knob, Pennsylvania is located halfway between Asheville, North Carolina and Burlington, Vermont. There is approximately 963 miles between Asheville and Burlington.
Your mom is there
One of the halfway points between Asheville, North Carolina and Carrollton, Georgia is Pleasant Retreat, GA.
Marion, IL
Consider Pittsburgh, PA.
Asheville,NC
Harrisonburg and Staunton, Virginia
Tuckers crossroads comes close to being the halfway point.
Asheville NC
Lexington KY and Louisville KY with Louisville being closer to the halfway point.
Interstate 80, Big Springs, NE 69122
The city between Asheville, NC and Myrtle Beach, SC is Columbia, South Carolina which is only about six miles from the exact midpoint. However, the town that marks the exact halfway point is actually Saint Andrews, South Carolina.
Halfway is between Little Rock and Fort Smith, Arkansas, closer to Fort Smith. I'd break that trip up in thirds.