Under NH law, a person can be charged and prosecuted for Criminal Threatening if he threatens to commit a crime against a person with a purpose to terrorize any person. Person A can threaten to assault Person B in order to terrorize Person C. Person C can hear or read the threat and Person B may not know about the threat.
So to answer your question: yes, your son can be charged with criminal threatening based upon what his ex-girlfriend received as a text if his purpose was to terrorize her. However, it seems that the police do not have the complete story and he should consult an attorney about the situation especially if he has the follow up text to show police.
This question has been asked by police and prosecutors for many years. It is known as a third party threat, and the issue has not been resolved by the courts in a satisfactory fashion. One thing to keep in mind is that an important element that the prosecution needs to prove in its case is that the person making the threat did so with a purpose to terrorize, meaning that the trier of fact must conclude that the the threat was made with a purpose to cause extreme fear.
Because the threat you described was sent by text, the prosecution need not charge the crime as criminal threatening but as harassment, which is a criminal charge that covers situations where threats are made by text. The same quesiton about whether a third party threat can be charged lingers.
This case certainly needs an attorney to review the facts and circumstances, particularly where you have described a portion of the message not shown to the police.
Note: The above is for educational purposes only and does not constitute specific legal advice, nor does the information form an attorney-client relationship.