Auri has some power, but what form it has is ill defined and warped by her broken mind.
Auri is broken. That is very much evident. She attributes thoughts and feelings to inanimate objects and places. She treats those feelings as more important than her own. She shows strong signs of an obsessive disorder, arranging all these objects to "fit together" in a way only she can perceive, that can change from day to day. Her rationale behind this is linked to "how the world turns", as though she were in a great stopwatch and trying not to get crushed by the gears.
Naming power is linked to the true name, and true nature, of objects. It's a very easy link to make, that Auri has somehow tapped into everything on a large scale. There are many broken people in Haven that show it's possible. The saddest part of Auri, in my opinion, is that its also possible that she's not, that something mundane broke her will, that she has no better understanding of Names or the ruins she hides in than anything else.
Her great tragedy is that we the readers know its possible, in her world, that a bronze gear can be obstinate. I'd want her to be right about that, that all she suffers means something. But for her to spend a week mulling it over where it goes, along with every other piece she finds down there... she may be in tune with something, but she is also broken.
In fact, since she's usually very afraid of changing something too far out of its nature, I'd say she's not using Naming. Knowing a Name involves understanding a thing, which she may / may not have an innate ability on, but calling the Name, invoking it, is to bend it to your will, which Auri is loathe to do in the simplest of situations. She cant even salvage bed linens without something else balancing her internal scales.
Rothfuss has mentioned there are other powers, other magics. I'd almost believe she'd tapped into something worthy of Fae, but there's little to say either way.