Possible to give Airport wifi extender a custom wifi name?
You can change the base station name in AirPort Utility by following these steps:


Open AirPort Utility
Click on the AirPort of whose name you want to change
Click Enter Password and enter the password if necessary
If this isn't necessary, click Edit
Navigate to the Base Station tab of the menu
Edit the name, then hit Update


But in terms of changing the Wi-Fi name, this is not possible unless you give the AirPort its own network, instead of extending an existing one. It will use the same name as the source network, like you said.