For day trips, minimal travel time is obviously important. If you're not in Beijing, you will go to a wall near your location.
Next is the question of what kind of Great Wall experience you want. If your primary goal is to visit the biggest tourist attraction, then Badaling is your first choice. If you want to visit a less crowded location, read on.
If you want to see the Great Wall where there is substantial tourist support for your convenience, consider going to Mutianyu. It is close to Beijing and easy to reach.
If you want a slightly less touristy but still developed location, Jinshanling is a good choice. It's farther from Beijing than locations in Huairou, but it's worth the extra driving time. Jinshanling offers very impressive fortifications and beautiful scenery.
If you want a more remote location, there are many choices.
If you want a slightly more challenging location than Jinshanling, Huanghuacheng is a good choice. Its central area has been restored, so it's comfortable to climb.
For a fully unrestored location, consider going to the Wohushan Great Wall in Gubeikou.
If you want something more extreme than Wohushan, consider going to Jiankou. Jiankou has everything, from great scenery to amazing walls. But it also presents great danger.
For a walk from one location to another, the route from Jinshanling to Simatai is a good choice. Another relatively easy route to consider is from Lianhuachi to Mutianyu. Not a lot of people walk this stretch of the wall, even though it's close to Beijing and convenient to access. It won't take you too long to reach Mutianyu, where you can then climb out of the south window of tower number one and onto the unrestored spur wall which provides even more challenge. Walking from Gubeikou to Jinshanling is a very good choice, but this route does have some issues, such as the detour around the military area, and so preparation is required. Other good day hikes include Zhengbeilou to Mutianyu and Huanghuacheng to Xishuiyu.
If you want to see some older Ming Dynasty Great Wall which is built of rough stones rather than brick and cut stone, the wall between Shentangyu and Lianhuachi is a good choice for an original and challenging wall, and the Qinglongxia Great Wall at Dashuiyu is a good choice if you prefer to walk on a rebuilt stone wall. Hiking from Shentangyu to Lianhuachi is a very demanding option. The stones in the wall are often loose and can be tricky to walk. But this is a challenging and rewarding hike that takes a full day.
If you want to visit a non-Ming Dynasty wall in this short time period, the only realistic choice is to see the Northern Qi Great Walls at Gubeikou and Simatai.
If you're handicapped, you should go to the Badaling Great Wall. It is the only Great Wall location with wheelchair access facilities. From the main entrance, you can take a combination of ramps and an elevator to reach the surface of the wall, and from there you can proceed for a moderate distance before further progress is blocked by stairs.