Where to wait in Atlanta in the night?

The smoothest option is most likely a flight from ATL to Asheville later on April 5th. Seems to be about $260 one-way through Delta. I don't recommend waiting overnight at a bus station as it can be both unsafe as well as uncomfortable; waiting overnight at the airport will certainly be better, though still uncomfortable. Considering the extra waiting time (for and on the bus) plus the cost of a hotel, the flight is probably worth it.

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.

I took a Greyhound into Atlanta for my hike of the AT from Springer, and I wouldn't want to do that again. When I did arrive, I went straight to the nearest MARTA station and got the hell out as fast as I could. I arrived in the early afternoon and I didn't think it was a very nice neighborhood even then. (I have a friend who lives in Atlanta who gave me a ride from there to the USFS 42 trailhead.)

Now, consulting my handy Thru-Hikers' Companion...

I would contact Bluff Mountain Outfitters there in Hot Springs at +1 828 622 7162 about shuttle service from Asheville, Tri-Cities or Knoxville airports, and then have her fly from ATL to one of those (from $140 one way, or consider changing her existing flight booking). This will probably be safer than any other option. These guys are amazingly helpful. I had one of my debit cards mysteriously cancelled while out on the Trail and was able to have a replacement overnighted there; I hiked into town minutes after it arrived and was able to pick it up without any trouble.

You also should give some thought to the return trip. It might make more sense overall to change the flight destination and return from ATL to, e.g. TRI, which is more convenient to Hot Springs, and much more convenient to points farther north of Hot Springs, like Erwin, Roan Mountain and Damascus.