It is an interesting question, certainly some parents are invested in the idea that they can "nurture" their child into being a genius. My experience with my three kids is that they were all different from the instant they are born, both in temperament and in interests. For me I felt there was a strong genetic component in what they as children were willing to invest their time into, and things which they really didn't care about.

As a parent what I tried to foster was curiosity. Asking basic 'Why' questions and following those questions to where ever they led. The goal there was to help them experience the dopamine hit that I get when I figure something out, its a great motivator for learning, and to help develop their tools for learning things on their own.

Did that explain why my eldest invented a new language? No, she wanted to be able to swear without being penalized for it. So I think there is certainly a certain 'make my life better for me' component to it as well.

I was pretty obsessed with computers as a kid, and my parents did not have the means to help me buy a computer so I mostly tried to find other ways to get access. And to the extent that they could support that, they did (like giving me a ride to the University's remote job entry building.) What I'm trying to say is that there is also a resource component to it as well, I know and have mentored kids who would show up in stories like this if they had access to a moderate sized bank account to facilitate research.

Bottom line for me is that the two largest influence factors that I've seen have been natural curiosity, and a support network that encouraged that curiosity.