Polygraph examination results are generally inadmissible unless all parties, including your son's attorney, agree to its admissibility. Strategically, it might be a good idea for your son's attorney to hire a polygraph examiner on his own and for your son to submit to the examination. If the results are in the favor of your son, his attorney can present them to opposing counsel and the court and they will be admitted if everyone agrees. If the results are bad, they are protected under the attorney work product privilege and do not have to be disclosed. Polygraph examination is very controversial. Experienced examiners will say that it is more likely for someone to have a false positive (meaning it will look like they are lying when they are not) than it is to get a false negative (meaning they somehow beat the polygraph examination).
It sounds like your son's lawyer has given sound advice in recommending that your son not agree to a mutually agreed or court appointed polygraph examiner. Your son should, however, consider hiring his own examiner (through his attorney) in an attempt to exonerate himself.
THIS ANSWER IS NOT FORMAL LEGAL ADVICE. It is provided for your information based on your AVVO Inquiry. Formal legal advice can be provided after consultation with an attorney. Answering your question does not create an attorney-client relationship or require further consultation.