On this issue, we would like to quote the fatwa of Dr. Muzammil H. Siddiqi, in which he states:
Making someone a godfather or godparent is a Christian custom. As far as I know among Christians (especially in the West, although traditions differ from country to country and from denomination to denomination), a person is named as godfather/godparent of the child at the time of child’s baptism. Among some Christians a godparent makes profession of faith for the person being taken as godchild.
With this, it is clear that a Muslim cannot participate in this custom and should not make a Christian profession of faith, neither for himself nor for anyone else.
However, if by godfather one means a guardian or a supervisor of a child, then there is no prohibition for that. If a child from a non-Muslim family needs Muslim help or protection, then he/she should be provided this help as much as possible. Muslims can help any person of any religion who is in need. Allah says in the Qur’an: (If any among the polytheists seek your protection, grant it to him, so that he may hear the word of Allah; and then escort him to where he can be secure ) (At-Tawbah 9:6).
Muslims who have such non-Muslim children in their custody should take good care of them. Hopefully, these children will hear the word of Allah and accept the right path when they grow up.
When we take any children into custody, we should keep in mind that they are not our biological children and when they grow up they will not be mahram (forbidden in marriage) to other members of the family. The Muslim ladies and girls in the household will have to observe hijab in the presence of the boys when they become adults and similarly the men in the household should observe the Islamic rules of the non-mahram when these girls become mature.