What is the ruling on not fulfilling a promise of marriage to a girl because the family objects to her being divorced and having a child?

Fulfilling a promise is commendable (mustahabb) but not obligatory (wajib), according to the more preponderant opinion. Therefore, there is no sin upon you if you withdraw from marrying her. Try to convince your family, otherwise, look for someone else. You could also help her find a righteous husband. The fact that a woman is divorced or has children is not a religious impediment to marrying her, if she is a woman of faith.