How to address the person being recommended in a recommendation letter?
I have seen both used in recommendation letters, regardless of the strength of the recommendation letter itself. I would, myself, recommend Dr. John Doe / Dr. Doe for most professional relationships, except when it is clear that you have formed a close professional relationship (PhD student, or long-term collaborator or post-doc). In case you go for the surname, I would still use the full name at the first mention:


  Dr. John Doe was a student of mine at the University of X, where I teach Y, …. He later joined my research group as a Masters, then PhD student. John is an extremely bright student, …


(Of course, that's merely an example and not a good letter wording, so do not look at the wording itself, only the use of names…)