True field level security in SharePoint lists is tricky to implement out-of-the-box.
Since you are only worried about one column, I would use the following methodology.
For discussions sake, and to continue your Netflix comparison, we'll say your main list is 'Movies'. You could create a second list called 'MovieQueue' which would have 2 columns:
- a number field called 'Priority'
- a lookup column which ties that list item back to a single Movie.
Your 'Movies' list would retain it's original security, so it would only be updatable by those with the right permissions. Your 'MovieQueue' list items could be editable by a wider audience without danger of altering your original Movie list.
The one thing I haven't discussed yet is how to make sure each 'Movies' item gets a corresponding entry in the 'MovieQueue' list. I would have a simple workflow that gets fired on any create/update of the 'Movie' item. The workflow attempts to search the 'MovieQueue' list to see if an entry for the current movie exists. If it does not, it creates the entry.
For basic editing the priorities, you could use the 'edit in place' feature of your basic list view. If you really wanted a rich user experience, you could use jQuery along with SPServices to create a true drag and drop experience for your users.