My husbands first child lives out of state, He has never met her, tried to contact the mother and she will not give any information to where she is so we can make contact with the child. He wants to know if it would be best for her for him to give up his rights to her. She has never met him and doesn't know him except to be a stranger.

He wants to know if it would be best for her for him to give up his rights to her.

What makes him think he CAN give up his rights? There is this myth that a parent can just give up their rights. But its only a myth.

What has he done to seek visitation? Does he pay child support?

My husbands first child lives out of state, He has never met her, tried to contact the mother and she will not give any information to where she is so we can make contact with the child. He wants to know if it would be best for her for him to give up his rights to her. She has never met him and doesnt know him except to be a stranger.

... and that's the goal?
He may relinquish his rights/ it is allowed in 23 states/ but he will be responsible for child support=

From my understanding, while there are areas that allow relinquishment of rights, it still has to be approved by a court and courts are very reluctant to grant TPRs.