Terribly annoying Apple ID verification popup

Some things you can try if you haven't done them already. Do you have access to another WiFi network where you can try? That has worked for several people.

Try a restart.

Restart your iPhone, iPad, or iPod touch         

Try a forced restart.

Reset Device.      

If necessary

Hard reset.        

Do a backup and try a factory reset. Then restore from your backup.

Restore to Factory setting.      

Restore - If you see an error when you update or restore your iPhone, iPad, or iPod

Backup - Restore your device from an iCloud or iTunes backup.      

If she creates/uses a new Apple ID, she will lose access to some of her data and the purchases made with her existing Apple ID.

Apple ID - create a new.      

Apple ID – Create        

Apple ID - Could Not Create or Could Not Set Up iCloud message.    

When changing to the new ID, there may be some data loss, but she can minimize it. If syncing photos with iCloud, ensure Optimization is turned off in Settings/Your Name/iCloud/Photos and/or in Photos/Preferences/iCloud. Then make sure full resolution photos are on the device/computer. When signing out of iCloud, if asked whether you want to retain data, choose yes. Messages conversations will be lost if they are addressed to the ID. See the article below for other places that the old ID should be removed. See Winston Churchill's response in the below discussion. If possible, also set up Family Sharing  between the IDs to keep access to your apps/purchases. 

 How to switch Apple IDs on your iPhone or iPad.      

How to transfer all files from one iCloud account to another.       

What to do after you change your Apple ID or password - Apple Support.      

Family Sharing.         

Family Sharing - Set up.