Have your mother try the links below.
Apple ID - How to find your Apple An iCloud e-mail address is an Apple ID.
Apple ID - Find
Enter her Apple (iCloud) ID and click Forgot Password.
Apple ID - Manage
Apple ID – Forgot Password
If she is the original owner, take the device, the original purchase receipt with the serial number, and a photo ID to the nearest Apple Store.
Genius Bar Reservation US
or contact Apple ID Support. They will need to see the original purchase receipt/paperwork also.
Apple ID Support
Apple ID Support (2)
Apple Support by e-mail or chat Select your device, Battery, Power and Charging, then Power adapter not working to get a chat option.
Apple Support using Twitter
Or fill out and submit this form.