How can my husband's son out of wedlock get a US citizenship if he is not on the birth certificate?

Your question requires an attorney consultation. It is not a simple question that can be answered on this type of forum. There are many factors that would need to be considered and evaluated. I strongly suggest that your husband contact an experienced immigration attorney for a face-to-face consultation and give him/her all of the facts surrounding his situation. (S)he would then be in a better position to analyze his case, advise him of his options and, if possible, make sure that the necessary paperwork and documentation is presented so as not to delay the process.

He needs to show that he and his son had father/child relationship before the son became 21. He has to show that he provided some kind of financial support to the son, he can show son's school records if the father's name is on the records. He may have to get a DNA to prove that he is the biological father.