THE PROBLEM:
Recently I've run into the problem with my 3ds XL getting the 001-0502 error code when I tried to connect online for my Monster Hunter 4 Ultimate game. The problem also affects the 3ds XL friend list connection, as you are not able to go online with the friend list when this problem occurs. However, the 3ds XL is able to connect to the internet browser and Nintendo eShop fine. I looked on the internet for solutions, but no one seems to be giving a consistent answer.
One description of the problem and "fix" that I found on the internet was usually something like: "It just suddenly started working again after [varied lenght of time]." or "I reset my 3ds XL connection settings and turned off my router for a while, and after turning it back on and setting up connection it worked." From this, I believe I found out the real reason and fix for the problem.
THE FIX:
The one thing you have to change is the DNS. Do not set DNS to auto-obtain. Instead search the Nintendo support pages for something like "DNS manual setting". You'll get a page where Nintendo gives you a specific DNS to input. Currently, the DNS that Nintendo has on that page is, Primary DNS: 8.8.8.8.
Secondary DNS: 8.8.4.4.
After you enter it as written, it should show up on your 3ds like this: 008.008.008.008 and 008.008.004.004.
I believe this to be the main problem given what people have experienced. The magical "It worked after a while" or "It worked after I reset everything and my router" seem to indicate that the DNS auto-obtain has input the correct DNS after a while. I'm no expert in networking so I can't be completely sure and I can't explain why auto-btaining DNS does this. If someone has an explanation, feel free to add on.