My son's Xbox 360 profile won't connect to Xbox Live

According to this Xbox Support page:

To access your Xbox Live account and other Microsoft services, you need to know the email address and password you used to set up your Microsoft account. Xbox Support can’t provide this information, even if you know your gamertag.

But don't fret yet. Here is something you can try:

1. Press the Xbox symbol button (middle of the controller)
2. Navigate to Settings
3. Click on Account Management
4. Scroll to Account Security

(Source)

At this point, the email associated with the account should be displayed. This will at least get you the email address used but unfortunately does not get you the password. Getting the password can be near impossible, since password resets require you to have access to the email addresses services. You can try contacting the email provider and asking for a password reset, but its unlikely that they will not do this for security reasons.