I would typically say it like this:
She is the girl with the rubber band in her hair.
We can say “the”, as we are referring to a specific girl, and therefore a specific rubber band.
And I would say “in”, as to me “on” sort of sounds like the band is resting on top of her hair, as opposed to being integrated as part of her hairstyle.
In addition, following your comments:
You are right in that the band is around the hair, and not within it. I suppose that saying “around her hair” might make sense, only when we say “her hair”, we’re generally speaking of her full head of her and its general style. The band is kind of at the back, so that’s why “around” might sound odd.
With the regards to the ring example, it’s interesting to consider. I suppose if we think about the process of putting a ring on, it’s very much placing it “on” the finger. This isn’t really the case with a rubber hair band. You’d typically pull the hair through, perhaps several times, twisting the hair a bit, it’s much more of a merging type of process, so perhaps that’s why we say “in”. I suppose, also, if you imagine an invisible bubble around her hair/hairstyle, loosely speaking, the band is “in”/inside of that; and we could say this of anything: “Gum in one’s hair”, “the wind in one’s hair”, etc.
Another difference is that a finger is singular, whereas “hair” is a general collective term.
Typing “rubber band” into google, I noticed that the top two autocomplete results were “rubber band hair” and “rubber band in hair”. The top result for “rubber band on” was “rubber band on wrist”, with the word “hair” not appearing at all.