There seems to be a common theme here: use WEP/SHARED instead of WEP/OPEN.
I don't understand why this would matter. I use WEP/OPEN. It works fine.
In many cases, the problem is most likely that the user did not enter the WEP key correctly in the DS or Wii. Changing the security setting to SHARED just forced the user to enter the WEP key again (this time correctly), and then they connect just fine. The user then falsely attributes the change to SHARED as a fix of the problem.
One advantage of going to SHARED is that the error code makes more sense when you enter the WEP key incorrectly. If you use OPEN authentication, and you enter the wrong WEP key, it usually gives you an error saying that it could not obtain an IP address. If you use SHARED authentication, and you enter the wrong WEP key, the device should give an error code that tells you that the WEP key is wrong.