I have a couple of suggestions.

1. I think you can make it truly persistent by making use of wild cards (*) in the URLs. Similar to those used in adblock and other addons. So for example letting the user set the URLs *.youtube.com/* to show the name 'YouTube' will make more sense to a lot of users.

2. A toolbar button is good, but for users like me with a lot of addons it's note very useful. At least an option to show the Rename function in the right-click tab context menu would be the best way. This also would help us rename the tabs that are not in focus. So not need to switch tabs for renaming.