Look into Numenta's Hierarchical Temporal Memory (HTM) concept. This may be slightly off topic if the expectation is of "traditional" Neural Nets, but it is also an extremely promising avenue for Artificial Intelligence.
Although Numenta introduced HTM and its associated software platform, NuPIC, almost five years ago, the first commercial product based upon this technology was released (in beta) a few weeks ago by Vitamin D. It is called Vitamin D Video and essentially turns any webcam or IP camera into a sophisticated video monitoring system, recognizing classes of items (say persons vs. cats or other animals) in the video feed.
With the proper setup, this type of application could make for an interesting display at the Science Fair, one with much "curb appeal".
To wet your appetite or even get your feet wet with HTM technology you can download NuPIC and check its various sample applications. Chances are that you may find something that meets typical criteria of both geekness and coolness for science fairs.
Generally, HTMs aim at solving problems which are simple for humans but difficult for computers; such a statement is somewhat of a generic/applicable to Neural Nets, but HTMs take this to the "next level".
Although written in C (I think) NuPIC is typically interfaced in Python, which makes it a convenient test bed for simple yet sophisticated proofs of concept applications.