Even if the OP's mother remembers the correct Apple ID and Apple ID password, that iPhone 4 isn't going to be of much use – except, perhaps, as a portable music player.
The iPhone 4 (GSM) and iPhone 4 (CDMA) have 32-bit processors and 0.5 GB of RAM, and are incompatible with iOS 8 and later. This makes them obsolete as a platform for running apps. I do not know if they'll even be able to access the App Store.
They might not work too well for Web browsing either – considering that they haven't received any updates since June 2014 – ten and a half years ago.
These iPhones also have no support for 4G or 5G cellular bands. In the U.S. and several other parts of the world, carriers have shut down, or are in the process of shutting down, their 2G and 3G networks. That would make the iPhone 4 (GSM) and iPhone 4 (CDMA) useless as cellular phones.