YouTube has this nasty habit of defaulting to 360p after every browser restart. Actually, when you switch to the wide video view, the player will also up the resolution to 480p and set a cookie called 'wide'. This is done so that subsequent videos will also be displayed in wide view.
Problem is, the 'wide' cookie is set to expire at the end of the browser session. We can work around this by manually adding a cookie with the same name to the database Firefox keeps all its cookies. We also set the expiry date to 10 years from now.
(For a more straightforward way to do this, skip to the last paragraph.)
Of course, you need to replace '' with the name of your profile directory in '$HOME/.mozilla/firefox/'. Also, Firefox should not be running when you run the command, otherwise you'll get an error about the database being locked.
Tested with Firefox 3.6.13 on Arch Linux. Should work on other Linux distributions too.
For Firefox 4 and later, use the following:
Now that I think about it, a cookie editor add-on for Firefox would accomplish the same thing with less hassle, and additionally work on Windows. After switching to wide view once, just modify the cookie not to expire at the end of the session. :p