I called Customer Support, and the development team is looking at it. They said it is happening randomly. They labeled it BUG-000151089. It only started to happen after they did an update in June.
The work around I was provided, until they fix the bug, was to change the ownership of the web map to another in your organization, and make the edit, then change the ownership back. But if you don't have another account in your org, I'm not sure what could be done as a work around.
I suggest giving Esri a call and letting them know you are having this issue as well. When I called, it seemed they weren't aware of this issue yet. So the more people who let them know, the faster it hopefully gets fixed.