In most states, students who enroll in the public high school before they turn 18 are allowed to remain in high school until they complete graduation requirements or until the end of the school year in which they turn 21. This is very common for students in special ed and for immigrant students - especially refugees with disrupted educations. One caveat is that only minimum graduation requirements need to be met. Students don’t automatically get more time in order to fit in honors classes.
Each state sets its own policies, so if your state has a strong network of adult schools, and routinely sends everyone there the minute they turn 18, then it might be a tough sell to get an exemption so that your kid can finish up at a regular high school. Do investigate the adult school structure. In some states these are full-on high schools that offer standard graduation requirements (again, probably no honors options), in others they offer few physical classes but include portfolio evaluation systems that allow students to demonstrate equivalent knowledge for a few classes that they missed along with coordinating credit for online classes, and in still others they just offer GED exam prep. I teach Adult Basic Education and GED prep for a school district that offers only that option for students who didn’t finidh high school. I do not recommend it at all for students like your son who are almost half-way done. If he can find a way to complete a true high school education (even withiut honors sections) he will have a much more thorough grounding in the subject areas, and he will have a happier social life.