Zina (adultery or fornication) is forbidden in all the laws that Allaah revealed to His Messengers, and Islam approves of the marriages of followers of other religion who did not enter Islam subject to two conditions:

1 – That it was in accordance with their own laws

2 – That they do not refer to us for judgement concerning the marriage contract.

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said:

The companions of Maalik and al-Shaafa'i, and the companions of Ahmad such as al-Qaadi Abu Ya'la and Ibn 'Aqeel, and later scholars, said that reference concerning the marriage of a kaafir should be made to their own customs. Whatever they regard as a marriage among them, it is permissible to approve of if they become Muslim and refer to us for judgement, provided there is no impediment to this marriage. But if they believed that it was not a marriage, then it is not permissible to approve of it. End quote.

Majmoo' al-Fataawa (29/12).

Shaykh Muhammad ibn Saalih al-'Uthaymeen (may Allaah have mercy on him) said:

If the marriage is valid according to Islamic sharee'ah, then it is valid, but if it is invalid according to the dictates of Islamic sharee'ah, then they may approve of it subject to two conditions:

1 – That they regard it as valid according to their laws

2 – That they do not refer to us (for judgement).

If they do not believe that it is valid, then they must be separated, and if they refer to us for judgement, we must examine the case. If it is before the marriage contract then we must do the marriage contract according to our laws. If it is after the marriage contract, we must examine it further. If the woman was permissible at that time, then we may approve of it, and if she was not permissible then we must separate them. The evidence for these things is what happened when a kaafir became Muslim at the time of the Prophet (Peace And Blessings Of Allaah Be Upon Him). He approved of those couple who had married during the Jaahiliyyah and did not object to that. This indicates that things may be left as they are. End quote.

Al-Sharh al-Mumti' (12/239, 240).

With regard to zina and so-called relationships, all of that is invalid according to their laws and ours. It is the result of the misguidance in their lives, behaviour and customs.

Muslim (1700) narrated from al-Bara' ibn 'Aazib the story of the stoning of the two Jews who committed zina, and how, even when the Jews distorted the Torah and concealed that which Allaah had revealed, they did not regard zina as permissible, rather they changed the punishment for it and introduced flogging and blackening of the face with coal instead of stoning.

The Christians did likewise. In the Gospel of Matthew 19:18 it says: "Jesus [said], 'Do not murder, do not commit adultery, do not steal, do not give false testimony…'"

In the Gospel of Mark 10:19 and the Gospel of Luke 18:20 it says: "You know the commandments: 'Do not murder, do not commit adultery, do not steal, do not give false testimony'"

Hence we say: If these parents were married – even of that was according to the religion of Christianity or Judaism – then their marriage is approved and the daughter should be named after the father. But if the daughter was the result of an illicit relationship, then she should not be named after the zaani, rather she should be named after her mother, as she is at present.

In Islam the scholars are unanimously agreed that the illegitimate child should not be named after the zaani if the zaani does not ask for him to be named after him. Rather the majority of scholars said that he should not be named after him even if the zaani wants that.

It is not the issue – as mentioned in the question – of whether the zaani is still alive or not; rather the issue is that the relationship between them was not one of marriage, and the daughter was the result of that relationship.

Islam forbids attributing the child to anyone other than his father. Allaah says (interpretation of the meaning):

"Call them (adopted sons) by (the names of) their fathers, that is more just with Allaah. But if you know not their father's (names, call them) your brothers in Faith and Mawaaleekum (your freed slaves). And there is no sin on you concerning that in which you made a mistake, except in regard to what your hearts deliberately intend. And Allaah is Ever Oft-Forgiving, Most Merciful"

[al-Ahzaab 33:5]

It was narrated from Abu Dharr (may Allaah have mercy on him) that he heard the Prophet (Peace And Blessings Of Allaah Be Upon Him) say: "There is no man who knowingly attributes himself to someone other than his father but he has committed an act of kufr, and whoever claims to belong to people to whom he does not belong, let him take his place in Hell."

Narrated by al-Bukhaari (3317) and Muslim (61).

Al-Haafiz Ibn Hajar (may Allaah have mercy on him) said:

Some commentators said: The reason why this is described as kufr is that he is telling a lie against Allaah; it is as if he is saying: Allaah created me from the sperm of So and so, when that is not the case, because he was created from someone else.

Fath al-Baari (12/55):

The Messenger of Allaah (Peace And Blessings Of Allaah Be Upon Him) said: "One of the greatest of falsehoods is for a man attribute himself to someone other than his father." Narrated by al-Bukhaari (3318).

And the Prophet (Peace And Blessings Of Allaah Be Upon Him) said: "Whoever knowingly attributes himself to someone other than his father, Paradise will be forbidden to him." Narrated by al-Bukhaari (4072) and Muslim (63).

To sum up:

The illegitimate child – whether he is born to Muslim or non-Muslim parents – cannot be attributed to the zaani, rather he must be named after his mother. The situation of this new Muslim sister is correct. If she was not able to call herself after the man or the woman then she could – as a case of necessity – call herself by a name that is not specific to any known person; rather she may choose a name that consists of two or three names and call herself by that. It is not permissible for her to take her husband's name.

And Allaah knows best.