What name should the Muslim mention upon slaughtering the Udh-hiyah in case the father is dead? Is it permissible to mention the name of the children or the wife?

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

The name that should be mentioned upon slaughtering the Udh-hiyah – after the Name of Allaah The Almighty – is the name of the owner of the Udh-hiyah. If he is the slaughterer, he will say, "I am slaughtering it on my behalf." If the Muslim slaughters an Udh-hiyah of someone else, he will say, "I am slaughtering it on behalf of so-and-so." But this in not obligatory. It was narrated on the authority of Jaabir ibn ‘Abdullaah that he said: "I attended Al-Adh-ha day with the Prophet, sallallaahu ‘alayhi wa sallam, in the prayer place. When he, sallallaahu ‘alayhi wa sallam, finished his Khutbah, he stepped down the pulpit, brought a sheep and slaughtered it with his hand, saying, 'Bismillaah wallaahu Akbar (In the Name of Allaah; Allaah Is Greatest). I am slaughtering this on my behalf and on behalf of people who did not offer Udh-hiyah of my Ummah.’" [At-Tirmithi] [Al-Albaani: Saheeh]

If the father of the children died, they in principle should offer Udh-hiyah for themselves. This is because Udh-hiyah is in principle slaughtered on behalf of living people, not the dead. Therefore, the Muslim who wants to offer Udh-hiyah should say upon slaughtering it on his behalf and on behalf of his family, "I am slaughtering it on my behalf and on behalf of my family”.

Allaah Knows best.