Continually asking for password for non-existent id

I have a little screen popup asking me to sign in under a non-existent apple id. (See photo) The problem is that even when I touch cancel, it pops up again right away - not allowing me to use my phone. This actually first happened when I was setting my new phone up using the data from my old X. Applecare really couldn't figure it out and said it must be an app. However, this id was my son's at one point long, long ago (he doesn't even remember that it was). My son would not have purchased any apps for my phone. I suspected it stemmed from iTunes. That was born out today when I synced my music from my Mac to my phone. When I tried to open Music on my phone, the screen popped up again and again. I have rebooted, but that has not helped. I have thousands of songs in my library and there is no way to figure out which ones he might have purchased long ago. I just really want this to go away. Checking under forgot my password, apple says this id does not exist. Anyone have any ideas of what I can do?