So, I purchased and created an account for myself, and another for my son. When I created a LAN game, my son was unable to join, because it said that his CD-key was already in use on the server. Not even possible, since as I said, I purchased two accounts, and I have looked at the tab where the CD-keys are, and verified that they are not the same.

They don't make this very clear during installation, but you need to manually add your personal CD keys to the game for multi-player. In the Neverwinter Nights group on your start menu, you should see a utility called CDKey Changer Tool. Run that and enter your personal keys.

You'll also need to download and run the Premium Modules Re-installer from your library, so that the premium modules get refreshed with your new keys.

Thank you!

Please remember to mark Ryan333's answer as the solution. :)