The distance from Atlanta to Asheville is not inconsiderable. Is renting a car and driving there out of the question?
Like many U.S. bus stations, the Atlanta Greyhound terminal has few services and is not optimally located. The South Downtown neighborhood is not especially dangerous by Atlanta standards; however, it is seedy, and the area where the station is located is dominated by state and local government offices— an earlier name for the district was Government Walk. Very few people (other than vagrants) will be around after business hours, and there are few establishments to patronize. I would not recommend my own mother wait alone at the station or nearby for 8 hours.
If you are insistent on the bus option, your mother could take MARTA from the airport to the Peachtree Center station downtown, which is in the hotel district. It will be too late to visit most of the tourist destinations in the area, like the World of Coca-Cola or the CNN Center, but Georgia Aquarium is often open until 9pm (you should book tickets in advance). There are many restaurants and hotels in the area where she can wait out the duration, then take a taxi or car service to the bus station, which I would recommend over the train late at night.
I have no experience with luggage storage in Atlanta. There is a bag storage business downtown but I have absolutely no idea as to their convenience. The Aquarium has a coat check, but I do not know if they would accept luggage. She could always try the bell desk at one of the larger hotels, but I am not confident Atlanta hotels will accept bags from people who are not guests.
Of course, it is also possible that your mother may just want to get a hotel room somewhere and nap, after a long flight and before a long bus ride on which she will probably not get much sleep.