I was actually writing my post purely from memory and your request made me determined to find out just what episode it was that I was remembering. After [a wee while] of searching I can tell you that the episode is: 'Bliss'.
http://memory-alpha.wikia.com/wiki/Bliss_(episode)
Here is a direct quote from the page:
"On the bridge, Kim tries to remotely drop the force field, but Seven has Naomi enter a code to block the attempts. "
Yes, but Seven gave her precise instructions to view and follow, that allowed the desired action to happen. It's not as if Naomi figured it out on her own. From that standpoint, I find what she accomplished on the holodeck a bit more impressive.
I thought someone may say that however I find that argument...unconvincing. I agree that Seven helped her and she didn't do it all on her own but the fact still stands that a [7/8/9] year-old girl, with some instructions, managed to override controls from the bridge. I still find that staggeringly easy for a Federation starship.
It's not an argument, it's what actually was depicted, so I'm not sure how you can characterize it as unconvincing. As far as I can tell, none of what happened was because of Naomi's intuition or knowledge.
Honestly, it's not significant enough to take any further, but I think there's a difference between being shown exactly the sequence of controls to deploy and simply remembering the string and carrying out the action, as in the former, and actually figuring out how to manipulate a control system on her own, as Naomi appears to do in the latter instance.
At the same time, while I know it wasn't the point you were originally illustrating, I would certainly concede that Naomi did seem to be very intelligent, perceptive, and displayed an interesting complexity of character for one so young.
My biggest Trek peeve: James Tiberius Kirk, a swaggering, womanizing dunce that somehow managed to swagger his way into command of a space cruiser with guns. (and no, I'm not talking about the Pine version.)
Next: Captain Jean-Luc Picard, an effete pacifist snob who was at least civilized enough to leave the swaggering and womanizing to his oak tree of First Officer.
Next: Captain Katheryn Janeway, an utter fool who managed to strand her ship 70,000 lightyears from home because she couldn't figure out "Leave charges on the array to blow it up AFTER we leave."
Any episode of any show where a FTL spaceship has to travel through a deadly gas cloud because the crew can't figure out how to go over it or under it (you know, like any fifth grader who builds model rockets can work out.)
My favorite TNG character getting killed by a talking oil slick.
There's a bunch more, but those are the biggies.
Well, beyond the obvious conclusion that there wouldn't have been a series if Janeway had done that, it certainly could have been portrayed as what they
attempted to do, but something (mechanical failure, Kazon intervention, miscalculation by Tuvok, etc.) could have simply prevented its successful execution, ending with the same result. This certainly would have made the establishment of the show's premise more plausible, though it would have probably served to further negate the Maquis complement's animosity towards Janeway, which as a planned for plot driver was pretty much eviscerated by the network as it was.