If a child is the illegitimate offspring of two kaafirs, can he be named after the zaani?
Question

                I have read your answers regarding woman taking her husbands name and i understand that is is not acceptable. I would still like to now, if it would be possible for a converted woman in that case when the woman originally has her mothers name because her parents were not married when she was born, and it is not possible for her to take her fathers name, since he is not alive anymore.
Praise be to Allah.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.