How does a member change their display name?
In order to change their display name, a member must have permission to do so. On your site, only Super Admins can do that. You can change that by going to Permissions>>General, and scrolling down to the Miscellaneous section: "CAN CHANGE OWN DISPLAYED NAME" is what you are looking for. You can change it by clicking the "Update" button next to that permission.
It's a pretty common thing for sites to restrict that permission, but it might not apply in your community. Some sites have problems with people who misbehave in some fashion, and then change their display name.
For instance, I manage a public community where we sometimes have issues with members. Sometimes members will ask ME to change their display name, and I'm happy to do that if it's appropriate. As a Super Admin, you can update their display name in their profile.
There's an in-between option, as well: you can set the permission to allow members to change their display name, and then set a recipe so that you are notified when someone does that, in case you need to keep an eye on the situation.
Let me know if you have any questions, or want help with the recipe I mentioned.