Security - Role Maintenance - Security Options (this is in version 2021) - Library Master - Allow User to Change Password. In version 2020 it is on the Module Options Tab.

Thank you.  Already had that option checked on Module Options tab.  However, I haven't allowed any access to Library Master module on the Tasks tab (we're running 2019).  Does the role also need access to User Settings in Library Master under Tasks?  I don't want this role to be able to change others' user settings.

Not sure if this helps unravel the mystery, but I created a new user and a new role. The role has access to everything except User Settings. Sure enough, when the new user tries to run Change User Settings:

However, it can still change the password:

I'm still not finding the exact setting in Role Maintenance, though. I thought maybe it was the same permission that lets users get into the "Hidden" program menu (File - Run), but now I can't even find that setting..

Solved.  Something got out of sync.  Fixed by unchecking Allow User to Change Own Password in Module Options, Save - then re-checking Allow User to Change Own Password in Module Options, Save.  No issues since.  Very odd.