Per,
You are correct. Title is a field that you are discouraged from modifying as part of the basic content type so adding drop downs for that not feasible without changing that default field.
My recommendation is that you
- create a new content type based on document
- add a new title field (but it can’t be called Name or Title as those are reserved in SP)
- make that your drop down field and
- remove the title field from being required and from showing on the default form
The easiest thing to do however would be to again allow them to type in whatever title of the document they have and select another field called category, type, or something that contained your drop down. That way you don’t mess with the default content type. Title still has a relevant meaning to the user, and you still enforce a naming standard on that content type which helps with using the organizer.