Answer in the comments.
My answer: I believe that Harry Potter may have had a godmother. Considering Lily’s best friends are deceased, maybe they just don’t speak of it, because of that fact. I’m also thinking that he may have not had one, and only a godfather; Sirius Black, since he was always the closest to James. For godmother however, if he did have one, I’d assume Lily’s closest friend; Marlene McKinnon.