You can edit the display name on the intercom directory by going to "Intercom Settings" > Intercom Directory
Use the Filters to find the User/Resident that you are trying to edit the display name for. Click "Edit" button by the entry. If you want the Name and Display Name to be the same you can make the edit for the Name and it will mirror on the Display Name. If you want them to be different, uncheck the "Same as Name" box and then make the necessary changes on the "Display Name".
Make sure to click "Save" button to save your changes. You can review your changes to make sure all is good.