The one thing that makes one kid stand out from others is that they finished what they started. Another is that they moved from a smaller project to a slightly bigger and harder project then to a slightly longer and harder project. They also explored a field in depth. They developed a bag of tools along the way.
There are several analogies. Consider the sports star who spends 2-4 hours a night practicing and another 2-4 hours a night studying tapes and learning from books. Or, consider the pianist who spends 2-4 hours a day working on one long piece, breaking it down, then integrating it, working it over and over. Or the actor in a long play learning each scene, then adding to the performance night after night.
Expanding one's knowledge in a given area means to learn as much as you can about a subject, then when you find questions no one knows, you then spend the time to formulate your own answers. Parents and other adults can help by providing guidance on being organized, on developing an overall roadmap, setting up a place to work uninterrupted, help them to pace themselves and to provide patience, and help kids to find mentors and access to information, but ultimately the kids need the enthusiasm from inside.
Most of what passes for "science" in public schools is not science. Science is testing hypotheses. All the stuff taught has already been proved a long, long time ago. F=ma - duh. It is empty brain calories.
Kuhn had some thoughts on science that pertain to this discussion.
http://en.wikipedia.org/wiki/The_Structure_of_Scientific_RevolutionsReal research means disproving hundreds of ideas until you find one that might work. It is a struggle.
With today's internet, you do not need a lab. You just need a good pc and need to know how to program. You also need a good math foundation. Then you can do original work in many fields or collaborate on lots of open projects.
This field is literally exploding:
http://en.wikipedia.org/wiki/GPGPUThe basic tools to do science are programming, lots of math, and the ability to read a book and then do the problems. For instance, if I wanted to learn how to do something, I would go find some graduate level classes in that subject, look up the textbooks used, then work to understand everything in them by studying them. If I had some math issues, I would step back, find some texts on that math subject, master them, then move forward. I would also get some software programs on that subject and play with them to learn all I could. I would also find out if there were local interest groups that met or online forums or national or regional meetings. If you know your stuff and have a deep interest in it, then you will make connections and learn more.