I don't know what a "name change certificate" would be, but in order to get a certified copy a court order, you have to first file a case for change of name, and then get the order. This could probably be done in one case for everyone, but frankly, I suspect that there are a lot more facts we don't have here. The only time I have ever heard of an entire family changing their whole names is when the enter the witness protection program. Since you're in this forum, I'm assuming that's not your situation.
Each person in the family would be required to file a Petition with the local court to change his or her name. After filing the petition, you would set a hearing date. At the hearing, the judge would sign an order legally changing the name of the person. The cost of the filing fees alone will be around $250 each. The name change certificate is probably just the amended birth certificate that you get after the court orders the name change. It sounds like there is more to this story so I would suggest that attorney help might be needed.