“In the mean-time the solution to the problem of kids and the internet is simple. Parents need to get familiar with the technology themselves so they can help their kids navigate the digital world.”
This is pretty much the conclusions reached in a study by Sheffield University’s Burnett and Wilkinson Their sample of Year 6 children browsing in groups at school, alone and with their parents side-by-side found a strengthening of parent-child bonds and greater levels of social and educational attainment. At school were sites were locked behind a filter, the disability of child autonomy raised a lot of questions.

As for a web museum, excellent idea. There are plenty of online points: Internet Archive and Project Gutenberg spring to mind, that provide a starting point for this.