Update: The add-on is now on AMO! Check it out! Also, feedback is greatly appreciated!
So I went ahead and made a jetpack called "Copy Short URL" and it does what it sounds like:
On any webpage, you get a new item in the right click menu called "copy short URL". When you click it, the add-on looks for a canonical short URL exposed in the page header. Currently, a number of major websites expose their own short URLs for any entry on their webpages, among these: youtube ("youtu.be/..."), flickr ("flic.kr/..."), Arstechnica, Techcrunch, and many more. If, however, the site does not name its own short URL, the add-on automatically falls back to making a short URL using tinyurl.com. Either way, after a fraction of a second, you end up with a short URL in your clipboard, ready to be used in forum posts, tweets, or wherever else you please.
My add-on won the contest in the "most useful" category. The prize was an awesome jetpack sweatshirt:
Hope you find it useful!