Who is Tsuruya-san? What is she? On the surface, she's an energetic girl with impossibly long hair. But is there something more to this girl? Let's find out, starting with her interactions with other characters.
Tsuruya-san's best friend and classmate, as well as the local time traveler, Mikuru Asahina is the person whom Tsuruya interacts with the most, yet we don't know much about how close the two are in their friendship, and we never see the two in class together, save for a single scene in the Disappearance film. This is one of those cases where the most frequent interaction is not necessarily the most telling. Let's look at a couple of her other interactions.
Tsuruya seems to be on a rather friendly basis with the protagonist, Kyon, judging from her general tone when speaking to him, and how she addresses him as Kyon-kun, with -kun being an informal honorific. Then again, she's a generally friendly person.
Lastly, we'll look at her interactions with the female lead, Haruhi Suzumiya herself. Like with Kyon, Tsuruya addresses Suzumiya with a nickname, Haru-nyan, which implies a close friendship with Suzumiya. But, as we know, Suzumiya attracts the eccentric types...
Indeed, Tsuruya is eccentric. She's exceedingly cheerful and energetic, easily amused, and seems to be in on the whole "Suzumiya's God" thing that Itsuki Koizumi's group seems to believe, given that her family is stated to directly support The Agency. At the same time, she's very headstrong and independent, since she appears to live on her own. Heaven knows why she lives in a massive mansion with a vacation home in the mountains.
If one word could describe Tsuruya-san, it would be "enigma". Her true identity, if not human, isn't really touched on much, but she knows more about the supernatural than she lets on. It could be that the way she acts is a cover for an ulterior motive, but we may never know. After all, the novel series appears to be completed, so if her identity was not touch upon in the 10th novel, she's human with knowledge. But she's nonetheless an enigma in a world filled with enigmas.