From what I understand of her description, she is not making a graphene supercapacitor. Based on her description, her device significantly outperforms other supercapacitors, and has comparable energy densities to classichal batteries. Given that this summary is what she actually submitted to the science fair, I doubt she would have won if it did not actually have merit to it.
Googling "Eesha Khare" took me to a bunch of crap articles saying only that she won the Intel Science fair and would revolutionize batteries. Intel's winner's page provided no additionaly details, and no link to her actual work. Eventually, I was able to find my way to the list of winners, which was nice enough to mention the title of her project. Googling the title finally got me to the document listed.
Even using the title in Google, while it did bring up articles that were nice enough to mention the name of her project, still did not bring up a single article that bothered to even link to her work.
This concept is not newly developed by this young student, it seems that she's just a summer student who used the labs work to submit to the Intel competition, this is actually a very common practice for high school students. As usual the media overstated her role, the work was not done in the 2.5 months she has off for the summer solely by her. That's not to say she might not be bright, but it's not her genius that developed this concept, although she may have contributed. In fact this work was first submitted for publication in 2010, long before she joined the lab in 2012.
http://www.usc.edu/CSSF//History/2013/Projects/S0912.pdf [credit: gizmo686]
Edit: Added reference showing she was not part of the Li lab in 2011. 
Why is it harmful, you may ask? Because every smart kid who didn't get handed a professional project to submit, and doesn't know the game, thinks that he/she is so much less smart/hardworking/creative/accomplished than these supposed geniuses, and is discouraged from continuing in science research.
Lab experience is great for kids, and we should be expanding that experience to everyone, not using it as a tool to cement the position of the upper classes.
There is lots of promise in this field, but I doubt that this young lady has jumped out in front of the pack.
That being said, I would be delighted if she made me eat my words.
That being said, I would also like to eat my words here :)
So her supercapacitor is roughly on par with similar nanostructured supercapacitors being developed. I must stress that the highest specific energy does not always mean the best technology, but practical/fabrication considerations can lead to different things making sense for different applications, in particular you have to watch out for the nasty tendency of some capacitors to short-circuit, which at supercapacitor energies means "kaboom!".
That last bit is why expensive tantalum capacitors are often preferred over cheaper fabrications with higher specific energy: tantalum is safe and reliable, which matters a lot.
edit: I should also point out that neither her supercapacitor nor other current nanostructured supercapacitor electrodes achieves an energy density on par with a cellphone battery. The specific energy of a lithium-ion polymer battery used in cellphones is roughly 150-200 Wh/kg: nearly ten times as much as this girl's claim!
"Andraka's sensor costs $0.03 (to compare to a $800 cost of a standard test) and 10 tests can be performed per strip, taking 5 minutes each. The method is 168 times faster, 26,667 times less expensive, and 400 times more sensitive than ELISA, and 25% to 50% more accurate than the CA19-9 test"
Discussion on Hacker News from last year: https://news.ycombinator.com/item?id=4265830
(Sorry, couldn't resist after seeing the remark on Gauss)
There was a very interesting article about them on HN last year IIRC.