Follow George's advice and get a lawyer.  It may be expensive, but in the long run it is well worth the money.  I would also suggest that once this is resolved and if your son is truly found to not have been implicated in what he was accused of that he file a suit against the previous employer and milk them dry.   

First, I think your son should consult a lawyer with experience in labor law to discuss his legal options. Martindale's lawyer locator service might help in finding one near you:

http://www.martindale.com

Does your son have a right to an administrative hearing prior to termination? Two relevant documents he might offer in rebuttal of the polygraph "evidence" are Dr. William G. Iacono's article, "Forensic 'Lie Detection': Procedures Without Scientific Basis":

http://antipolygraph.org/articles/article-018.shtml

and Dr. Drew C. Richardson's "Evaluation and Opinion of CQT Polygraphy":

http://antipolygraph.org/articles/article-028.shtml

The fact that your son was interrogated at length about the relevant issue prior to the polygraph would have the expected effect of sensitizing him to the relevant questions, making a false positive outcome more likely.

In August of 2005 my son passed a polygraph in order to be accepted to the police academy. The examiner said he barely passed. He said it showed deceptive when asked his name. He was asked that many times. Last week he was called out of class and told he needed to go to Internal Affairs the next morning. He was told it had something to do with his last job. He called me very upset, because he had left in good terms with last employer of over a year. We could not imagine what the problem was. He was questioned over two hours the next morning over an "rumor" that had surffaced. Some supplies had been ordered in a higher quantity than was supposed to be and had to be sent back. This happened about the time my son left and they assumed it was him. This is a rather small town and somehow this got to the police department. He told he did remember the incident and if he did do it, it was by mistake. He liked where he worked and everyone there. What did he have to gain? They told him he would have to take another polygraph to prove he wasn't lying, that afternoon. He called me at noon, he was sick to his stomach. He could not belive this was happening!Long -story-short - he failed the polygraph. Now it looks like they are going to let him go.  We are so upset and don't know what to do.  He is 28 and this was going to be his career! He wanted to make a difference and now he won't even have a job. I don't belive he should have taken the test as he had just been interrogated and was upset. Also when he took the second poly he asked the examiner(same as first) why it showed he lied about his name on the first, he said he never said it. Infact there were many things that happened that did not seem right. Where do we turn?????