Can someone remotely change my mother's iPhone passcode?

For a normal personal iPhone, changing the device passcode generally requires access to the phone itself. An Apple account password and the code used to unlock the phone are different things. Changing one does not automatically change the other.

Account password changes or recovery attempts can generate security notifications, but I would not treat the absence of a notification as proof that nothing happened. Check the account's devices and trusted contact details with her, and contact Apple through its official support channels if anything looks unfamiliar.

Frequent resets can make it really hard to keep track of which code is current. With her agreement, write down the current phone passcode and Apple account password separately and keep them somewhere secure. Then check which one the screen is actually asking for before resetting anything.

She does not have to hand over control of her accounts for someone to help her keep a record.

Since this has continued across multiple devices and accounts, a patient walkthrough may be more useful than another new phone. A local technology class or support session could help her get comfortable recognizing the different prompts and entering the right credentials.

If the suspicion or difficulty remembering codes is new, or if similar problems are showing up elsewhere in daily life, a conversation with her doctor could be worthwhile. There are many possible explanations, and a phone problem alone cannot establish a diagnosis. Try to approach it as concern for her well-being rather than an argument over whether she is right.