Before you turned it off, did you investigate your son's Contacts? If he added a new contact or edited an existing contact did this also get added or changed to your contacts? If so, then his and your Contacts would seem to be signed in to the same iCloud account. It is possible to have the iPhone signed into his iCloud account in Settings > [His Name] and yet be drawing the data from another account using a different Apple ID. See: Settings > Contacts > Accounts. Indeed, at that setting you can also get the phone to sync data from other sources such as a Google account, MS Exchange, Yahoo, and others.
However, it is more likely that he has contact data from his own account but which was originally yours. You said this was a new iPhone. Did he have any other Apple device using that Apple ID with which he shared contacts?
Some setting, perhaps in the past, has connected your data to his account. You need to work out if this connection is still active, or if what he has now is simply a legacy of a former connection and he can proceed happily to delete the content he no longer needs without risking your own data.