I'm not really sure what area of research this would fall under. The closest I can think of is "infantile amnesia," although that would be a stretch. As far as I know there is no established way to classify ideas, so that would make this a question not very well-suited for academic research. Actually, one of the reasons why I asked is because I'm interested in looking for patterns in what people consider to be interesting ideas. So far there are a lot of parallels with the answers that people here have been giving for the "what have you discovered" question, which is amusing, albeit not especially surprising.

If you are interested in infantile amnesia though, this paper is kind of interesting and gives a decent overview of the different theories:

Harley, K. & Reese, E. (1999). Origins of autobiographical memory. Developmental Psychology, 35, 5, 1338-1348.

According to my understanding of Piaget's developmental stages, the earliest a child can have "inventive" thought is in the concrete operational stage. Children enter this stage during school years, but before puberty. It's during this time that children begin to grasp basic scientific concepts like conservation of volume.

However, I didn't pay all that much attention in developmental psych, my textbook isn't handy, and the Wikipedia article is not all that helpful. :)

Researchers have found that most of Piaget's milestones actually happen much earlier than he had thought. Basically new methodologies have been invented to simplify the questions so that children can show competence at an earlier age. Before many children had been failing to perform at things they were able to do because the tasks involved ancillary things they were incapable of, for example they relied on background knowledge that wasn't there or unrelated motor skills that hadn't yet developed. I'd be interested in seeing what Piaget's methodology was compared to the current state of the art in terms of measuring inventive ideas.