A private apartment can impose whatever requirements or restrictions they desire on applications as long as those applications do not violate any federal laws.  Federal laws prohibit discrimination on the basis of sex, religion, race, or some other types of protected classes.  Having a DUI is not automatically considered a protected class.  If you can show that the DUI is really a cover for a violation of federal law, then your son may have a discrimination suit.  Without some evidence that the excuse is fake, then your son may have to simply look for another place to live.