A button to generate a shortened link. For example, when you click share on YouTube, you get a youtu.be link.
For example, youtube will often include extra garbage parameters for stuff like playlists - which the youtu.be shorturls omit, making them perfect for sharing videos.
You could either provide some sort of button to your browser to place this into the clipboard, or use an extension like pentadactyl to add a keyboard shortcut for it. (It's simply “y” by default - so copying URLs works like yanking in vim)