Don't change your ID. Have your cousin use the links below to try to recover their ID. If the cousin isn't successful, have them contact Apple ID Support.
Apple ID - How to find your Apple An iCloud e-mail address is an Apple ID.
Enter your Apple (iCloud) ID and click Forgot Password.
If necessary, there is a contact link.
Apple Support by e-mail or chat Select your device, Battery, Power and Charging, then Power adapter not working to get a chat option. If you need to talk to them, ask them if they will do a Skype or FaceTime with you.
Or fill out and submit this form.