They gave him some paperwork, he should hang on to it because it has his 'reference' number on it.
Once your son is stateside, he should download and read the guidance and then create an account at the Visa4UK portal.
It's an optional step, but he can get a Subject Access Request from the UK government to see if they recorded him as having engaged in deception or something similar.
When he's ready, he should go ahead and complete the application and send it to the British Consulate General in NYC. They will issue a decision in about 5 - 10 days and let him know. If he's successful, there will be a bright, shiny entry clearance in his passport and when he arrives in the UK, his landing interview will be reduced to a brief and pleasant formality. If he's not successful, then you can take things from there.
All of what I have described above is a 'best practices' solution. Your son also has the option of presenting himself at another port of entry and seeking admission on his US passport. The downside to this option is that if he gets caught, he may attract a ban and then neither option will work.
That gives you two alternatives.
Approaching the US State Department for intervention isn't going to work for your son. This sort of thing happens all the time and at best they will give you a link to their web site. You can try instructing a UK solicitor with a practice area in removals, that will definitely help, but it will be expensive.
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