So let me see if I get this straight, your son's netbook (its called a Netbook, not a Laptop due to its small size and limited functionality) can only connect to the internet through the little operating system that boots before windows (I think HP uses the Splashtop OS), but when Windows itself boots you cannot access the internet through wired or wireless connections?
You say it has worked before? When? Has anything changed, such as updating drivers or installing software?
Go into device manager, and see if there are any problems for the wire and wireless connections. PC Jargon » Device Manager
This page will instruct you on accessing the Device Manager once you are in Windows.
Let me know if you encounter any problems, and what software has been installed or uninstalled recently.