I don't read the LN so I don't really know what she really is. She seems to have great power. She can make things happen just by thinking about it. She even create a parallel world. And she seems very dangerous even when she's unconscious. So, what exactly is Haruhi? What is her power? Did they explain it on light novel?
As described in Wikipedia, Haruhi possesses unconscious god-like abilities to change, destroy, and reshape reality to her desires. In an interview, Tanigawa stated that the idea for the character came during a sleepless night at the beginning of the 21st century.
I think Koizumi also mentioned that Haruhi has a God-like power. I just don't remember what specific episode did he mention it. Along with Koizumi, Yuki and Mikuru revealed to Kyon that that was the reason why they were sent to earth, to watch over Haruhi since she is unaware of her God-like powers.
You're not meant to "know" what Haruhi's power is, but instead you're meant to draw your own conclusion about what Haruhi "is"; which is one of greatest charms of this show. Koizumi, Mikuru, and Yuki all have different conflicting views on Haruhi, and the viewer is meant to have one of their own; whether they can draw their own original view on Haruhi or side with God/TimeParadox/SuperHumanEvolution theories the show offers is up to the viewer themselves.