I have a picture taken by a friend of mine that I wish to use on a book cover. Normally, credit would be given to the photographer with a line on the copyright page of the book with all the other copyright information. This photographer has expressly given their consent to me to use their photo but wishes that their name not be used. What's the proper thing to do?
Keep it simple.
I thank an anonymous photographer for permission to use the photo on the cover of this book.
The typical citation would be:
Reproduced with permission.
If the author doesn't want you to name them, then simply don't.
The person could always register a pseudonym or alias with the local or regional (state, etc) government and use that for any forward-facing identification.
If they don't want to go that far, they could always just give you a nickname to use.
I think that the right thing to do would fulfill the person wishes and say that I thank an anonymous photographer for the use of this photo On my book.