Things like passwords need to be kept with important records. They are no longer simply a way to post to a bulletin board chat. People put their entire lives on mobile devices these days. They now can mean real money, such as potentially that iPhone could be useless to your mother because of new security features. Your mother will either have to recover the account information, or she will at least hope she has kept the receipt from when she purchased to device.
● Read this article to learn how to remove Activation Lock either on the device itself or on the web --> How to remove Activation Lock - Apple Support
● Help with a lost Apple ID or password:
• For help with a forgotten Apple ID (this is likely an active email address or one used in the past), click here --> If you forgot your Apple ID - Apple Support
• For help with a forgotten Apple ID password, click here --> If you forgot your Apple ID password - Apple Support
● Reset device if you can prove to Apple's satisfaction you are the owner:
Apple can completely reset a device if a person can provide sufficient evidence of valid ownership. This is done online* at: Activation Lock support request - https://al-support.apple.com/#/additional-support Apple says, "Proof of ownership documentation is required," but does not provide many details in the article as to what is acceptable proof. Prior to this official service Apple typically required the original receipt** issued by Apple or an Apple Authorized Reseller and documented provenance (so typically it would be the first owner who does this, or a descendant with legal proof of ownership transfer). Note that while this will let you get use of the device again, it will not help you with lost access to an account.
* There is a special link for this service. To save time you should use it instead of https://getsupport.apple.com/