I have 2 360s. The first one (launch unit), I used to register my gamertag and I never downloaded any games on that one. Then I got a second 360 (launch elite), and put my memory card in that one to use my gamertag. Every time I download a game on my new 360, it still registers the download to the first 360 that I made my gamertag on. I have to be logged on to XBL on my elite in order to use any downloads.
Whichever 360 you made your gamertag on is the one that every download you make, regardless which 360, gets registered to. You'll have to migrate your son's gamertag to the new 360 in order to have that 360 the home console for that gamertag.
Edit:
You can download any demo on XBL on any 360 that isn't yours. As long as you use the gamertag that you bought the game with, you can always play it as a full game on other consoles, as long as your logged on XBL.
