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.

Beyond co-authors or people making a significant gratis contribution (e.g.: proof-reading your manuscript for free; helping you with translating between languages for free), nobody is entitled to a free copy of your book. You should disburse your complimentary copies however you want.

Personally, speaking as somebody about to complete a PhD myself, my priority will be to give a complimentary copy to my supervisor, because he has been outstandingly supportive and, to be honest, I have a bit of a grudge against my department (for various reasons which I do not wish to enumerate).

If you are concerned about who will benefit most, keep in mind that retired academics are often still actively interested in recent research, may not be very wealthy (i.e.: are less likely to be able to afford your book than a department, which should have a budget for such things), and may really appreciate being kept in the loop.