Islam forbids women to marry non-Muslims. This is something on which there is consensus among the scholars, with no difference of opinion on this ruling.
Al-Qurtubi (may Allaah have mercy on him) said:
The ummah is unanimously agreed that a mushrik cannot marry a believing woman, because that is, in a sense, a humiliation of Islam.
Tafseer al-Qurtubi, 3/72.
It is permissible for a Muslim woman to marry a Muslim whom Allaah has guided to Islam after he was a kaafir, and it does not matter if his family is Catholic or follows some other religion. It also does not matter if he became Muslim a long time ago or if he entered Islam recently, but what does matter – and is of the utmost importance – is that his Islam is genuine and he is not pretending to be a Muslim so that he can marry a Muslim woman, as there have been cases where men pretended to have entered Islam on the basis of conviction when that was not the case. If this is found out to be the case, then such a person is not to be treated as a Muslim.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him) said:
It is essential to understand the Kalimah which is Laa ilaaha ill-Allaah. It is the best of speech and is the basis of Islam. It is the first thing that the Messengers (peace and blessings be upon them) taught to their people and it is the first thing that the Messenger (peace and blessings of Allaah be upon him) taught to his people when he said to them: “Say Laa ilaaha ill-Allaah and you will succeed.” Allaah says (interpretation of the meaning):
“And We did not send any Messenger before you (O Muhammad صلى الله عليه وسلم) but We revealed to him (saying): Laa ilaaha illa Ana [none has the right to be worshipped but I (Allaah)], so worship Me (Alone and none else)”
Every Messenger said to his people: “Worship Allaah, you have no other god but Him.” So it is the foundation of Islam, and it is essential for the one who says it to understand what it means. It means that there is none who is rightfully worshipped except Allaah. There are conditions for that, which are: knowledge of what it means, certainty and absence of doubt that it is true, sincerity towards Allaah alone, belief in one’s heart and on one's lips, love for that which it indicates of sincerity towards Allaah, acceptance of that, submission to Him, belief in His Oneness, disavowal of shirk and avoidance of worshipping anything but Him, and belief that such worship is false. These are all conditions of saying Laa ilaaha ill-Allaah. It is said by the believing man and woman whilst rejecting worship of anything other than Allaah, submitting to the truth and accepting it, loving Allaah, affirming His Oneness, being sincere towards Him, and not doubting its true meaning. For some people say it but they do not believe in it, like the hypocrites who say it whilst doubting it or disbelieving in it.
So it is essential to know, have certain faith, believe in it, be sincere, love, submit, accept and disavow other gods.
Fataawa al-Shaykh Ibn Baaz (3/49, 50).
After the Muslim woman marries a Muslim man, their children are to be named after their father, and it is not permissible to do otherwise, even if his family are kaafirs. Many rulings are connected to this attributed, such as kinship, inheritance, and whether marriage is permissible or not, and so on. Hence it is not permissible to name a Muslim child after anyone but his father and family. There are texts which speak in strong terms about this issue and state that the one who goes against that is committing a major sin.
It was narrated that Sa’d ibn Abi Waqqaas and Abu Bakrah said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Whoever claims after having become Muslim to belong to someone who is not his father, knowing that he is not his father, Paradise will be forbidden to him.” Narrated by al-Bukhaari (4072) and Muslim (63).
And it was narrated from Abu Dharr (may Allaah be pleased with him) that he heard the Prophet (peace and blessings of Allaah be upon him) say: “Any man who knowingly attributes himself to someone other than his father has committed an act of kufr. Whoever claims to belong to a 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:
This hadeeth shows that it is haraam to deny a parentage that is known and to claim to belong to someone else.
Fath al-Baari (10/308).
It is not known that anyone among the Prophets, Sahaabah, Taabi’een or scholars changed their names because of their fathers or grandfathers being kaafirs. No wise man would do that, because of the evil that results from that.
Anyone who studies the books of biography will find foreign names of fathers and grandfathers for many of the Muslim scholars, when Allaah blessed the sons with guidance to Islam whilst their families remained kaafirs. These scholars did not change their names even though the names were foreign and their people followed kaafir religions.
When Islam forbade adoption, it also forbade changing the name of the adoptee’s father and tribe. Allaah says (interpretation of the meaning):
“Call them (adopted sons) by (the names of) their fathers, that is more just with Allaah”
In a case where the father is not known because the child is a foundling – for example – he should not be attributed to or named after any specific person, rather he should be called as a brother or mawla (freed slave), because Allah says at the end of the verse quoted above:
“But if you know not their father’s (names, call them) your brothers in Faith and Mawaaleekum (your freed slaves)”
One of the reprehensible matters in which some Muslims follow the kuffaar is giving the wife her husband’s name. This is haraam. Rather she should keep her father’s name.
It is permissible for a Muslim woman to marry a Muslim who has recently converted out of conviction and sincere belief. And it does not matter if his family follow any kaafir religion. The children should be named after their Muslim father and his forebears even if they are kaafirs, and it is not permissible to do otherwise.
And Allaah knows best.