Quick caveat: the narrator only breaks into the first person five tiny times in the whole book. Blink and you might miss it. Usually, when he does it, it's to imply something saucy. For example, at the very beginning he gives us the old nudge-nudge, wink-wink about Winterbourne's older lover in Geneva:
What I should say is, simply, that when certain persons spoke of him they affirmed that the reason of his spending so much time at Geneva was that he was extremely devoted to a lady who lived there—a foreign lady—a person older than himself (1.2).
It's like saying, "I'm trying to say a lot of nice stuff about him, but I'll break this third-person routine to tell you that people gossip about his love life."
Unlike other first person peripheral narrators (like those in The Great Gatsby or Heart of Darkness), we know absolutely nothing about the guy. We're talking a first-person narrator so peripheral, most readers do not even recall the "I." This little recognized, but much discussed, feature of the text makes it an excellent focus for essays or discussion questions. Why does James include the "I" at all?
Seriously, we're asking.
To further complicate things, though this book is all about Daisy (the title at least would lead us to think so, right?) we only ever really see Daisy through Winterbourne's eyes. His opinions cloud every aspect of the book, so the narrator's judgment is hardly an objective one, but rather just a means of recounting Winterbourne's impressions of others.
This makes the whole novel read like a bit of a game of telephone: Is Daisy really like that? Or is that just what Winterbourne thinks? Or what the narrator thinks Winterbourne thinks? When you really let yourself get into this train of thought, Daisy Miller starts to feel like Inception with tea parties.