Overnight stay in Birmingham?

Oak Mountain State Park

Cheaha State Park is right off I-20 about an hour past Birmingham. Closer to Bham than ATL. Nice park and campground, but it is a steep drive up to the campground. Also in this area is a Camping World with a little campground, but it is right on the interstate, would be fine for a couple of hours of sleep, but for anything more, I'd drive up to Cheaha.

If you are just looking for a place to rest for a bit, the City of Hoover (10 miles south of Bham) has a campground at their stadium. It's basically hookups in the stadium parking lot, but close to resturants and shopping.

Oak Mountain is nice, but will be a bit out of the way going to ATL, great option if you were going south.

Oak Mountain SP is south of Birmingham off of 65. We camp in the spring each year and really enjoy it.

Mt Cheaha is also beautiful located about 20 minutes off of I20 just east of Oxford, AL, but is not a drive for the faint of heart. For a quick overnight, as said before, CW has a campground right next to their location in Oxford. The interstate sits next to if and a RR track about a half mile away. Lots of great restaurants nearby.

There is also Lakeside Landing in Pell City off of 231. It's about 8 miles south of I20 right on the lake. Good restraints nearby. Easy access but tight spots.

Of the three I'd try Lakeside Landing for easy of access, quiet and not far off of I20.

We like to stay at the Hoover rv park at the Hoover mets stadium when we are passing through.

It looks to me that it is about a 3 1/2 hr drive to Birmingham from Memphis. To get to Oak Mt. or Hoover park will take you 30 minutes or more (I think probably a hour because of road construction and traffic) off your intended route. so you can be a long way down the road in that time. The CW camp ground in Oxford is nothing fancy but is easy on and off I20.
So how many hours do you want to drive in one day? If you don't mind me asking what part of Atlanta are you going?
There is a small camp ground just before you get to Birmingham I know nothing about. Five Mile Creek Canoe and Camping

Stayed in a Campground in Calera Al, ran by a Church group back in 2010!

If can boondock, there is a Bass Proshop at the Leeds Exit on I20 that has RV parking and a big gravel lot on the NE side of the parking lot. There is a nice outlet mall just 1 minute away for shopping and restaurants nearby.

we stayed at the KOA South of Birmingham.
Birmingham South KOA

H E R E - Guin area is a possibility, that is actually in line with what you asked...close to I-22 or I-20. And would put you close to equidistant between Memphis and Atlanta. Since you're just looking for an overnight stop, this should work for you. Not a whole lot in that area. I had family in the Jasper to Sulligent general area, there's just not a lot to offer, unless you want to get really far off the interstate in the Bankhead NF.

AL Exit 287 is Rickwoods Cavern SP. It is north of Birmingham. We stayed there on a trip to Florida. There are about 12 sites, cave, pool, gated.