Thanks for all the suggestions. Using a quote from a movie is cool but not really appealing to me. I think I'm going to stick with Broken Wing Productions until I find something better.
Also, can I use a song name as a company name?
He means just what he said; the name of your production company should have a reference in some way to the film industry. "Bad Hat Harry Productions" is one of the producers of the TV show "House".
This is their vanity card:
https://www.youtube.com/watch?v=rT5TAy7hJ_8&NR=1
It is a reference to this scene in the film "Jaws".
https://www.youtube.com/watch?v=zSo5fNkNKlw
His suggestion is to pull something obscure from your favorite or other film for the name of your production company.