We also ran into this problem for one of our users following an upgrade to ArcGIS Desktop 10.3.1. After some trial and error, and then opening a support case with ESRI, we found this solution:
Opened Regedit, then browsed to key: HKEY_CURRENT_USER\Software\ESRI\ArcGIS Online\SignIn\www.arcgis.com. For the affected user, there were entries there for AutoSign (value set to 1) and Password (value set to a series of hex numbers). No entry for Username was present. We deleted the AutoSign and Password entries from registry key. After then, the problem no longer occurred in ArcMap.
It would seem that this problem is due to a combination of
a) The ArcGIS Online sign-in-automatically setting is turned on
b) The user’s stored password is incorrect, or a password was stored but no userID
c) The ArcGIS Online sign-in dialog does not persist setting changes when the Cancel button is clicked. (We tried unchecking the option in the dialog for "Sign Me In Automatically" but it wouldn't stick following the Cancel button click.)