Stopping at the beach to "stop and stretch your legs" on the way from Atlanta to Orlando isn't really a "stop" on the way. Whether you head to the gulf beaches (Clearwater Beach as spiceycat suggested, and I concur), or to the Atlantic beaches, you'll add at least 3 hours to a 7.5 hour drive from Atlanta to Orlando.
If you want to hit Clearwater Beach, you'll need to continue south on I-75 south instead of taking the Florida Turnpike into Orlando. From this point you'd have about 60 miles to WDW if you'd take the Florida Turnpike. Its about 100 miles from the Florida Turnpike turnoff down I-75 to Clearwater Beach. Once you dip your toes in the ocean and get in the car to head back to Orlando on I-4, you've got another 100 miles. (I-4 is under construction between Tampa and Orlando, so you may run into construction delays, I've been lucky and haven't hit them, but others have!) So basically you've added 140 miles to your trip by stopping at Clearwater Beach.
If you want to hit the Atlantic beaches (Cocoa Beach is the closest), go on and take the Florida Turnpike into Orlando and then pick up Florida 528 going east. Its right about 70 miles into Cocoa Beach. Dip your toes in ocean, get in the car and head back the way you came, another 70 miles. So, basically you've added 140 miles to your trip by making a side trip to Cocoa Beach.
Whichever beach you hit its going to add pretty much the same mileage, and probably the same amount of time all things being equal. I've been to the beaches on both sides, and I definitely prefer the gulf coast beaches (Clearwater Beach). The gulf beaches have much gentler surf, clearer water and white sand. The atlantic beaches have rougher surf, dark water, and darker sand.
Whatever you decide, have a great trip!