How will failing a PhD qualifying exam affect admission to other PhD programs in the future?
Just because your son failed a qualifying exam for university X doesn't mean he's not qualified to get a PhD at all. 

The chances of this affecting future PhD admissions are slim as they would only consider the current qualification and skill-set of the candidate.  


  ...you didn't fail, they failed you.


This reminds me about Thomas Edison Mother's Letter. You said the right thing to your son! Because, more than anything else, he shouldn't lose his self-confidence. 

You believe in your son and so does his MS advisor. However, whether he has the potential or not, talk to him and find out whether he has the passion for research. PhD requires more of perseverance than skill. 

If he still got the interest, then by all means encourage him to apply again to another university. 

My best wishes!