No difference... when I view the list of contact forms (there are only two) and hover the mouse over the one I want to change the name of ("Contact Form1_copy"), I get a tooltip nearby that says "Edit", and it shows the same destination link at the bottom of the browser as when I hover over the actual "Edit" link below the name of the contact form (next to the "Copy" link).
Is it a WP 3.6 issue, perhaps? This is on a fresh WP install so the WP version is the latest.
Thanks for trying - any other ideas out there on renaming a form? I don't even mind going into the database with phpmyadmin if it's not going to wreck anything else...