Actually, biological is unimportant. Most states require the consent, or at least notification, of the person who is on record as the child's father. Again, we need to know what state to advise further. If the husband who is not the biological father is currently presumed to be the child's father, that is the person you need to get permission from. If that's not the person you want to treat as the father, then you need a separate paternity/parental rights/adoption action. The courts generally take a dim view using a name change as a defacto attempt to change parentage.
It is possible to request a name change on the N-400 form when petitioning for citizenship. That will require that the citizenship hearing occur at a judicial ceremony. It is also possible for the judge to decline the request. Without considerably more information than we presently have, I would suggest obtaining a name change through the state courts, and then introducing the proof of the name change in order to seek a certificate of naturalization that matches the child's new name (assuming the name change was approved by the state court).