Etiquette - send a copy of my new book to the department I earned my PhD within or to my advisor?
To the department. The point of publishing work is to share the knowledge with others. This will be more greatly achieved in your previous department where other professors and up-and-coming students can have access to your work. This would be better than your work simply sitting on your old supervisor's book shelf. Your supervisor likely knows your work and therefore does not need a constant reminder sitting on his or her shelf. Also, I never knew an academic supervisor who cherished having a previous student's work sitting on their shelf solely for sentimental reasons, which I think might be your motive.