Can she marry a Muslim whose family is Catholic? Can the children be named after his kaafir people?

Firstly: 

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. 

See also the answers to
questions no. 69752,
6402 and
22468. 

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)”

[al-Anbiya’ 21:25].

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). 

For more details on the
conditions of bearing witness that there is no god but Allaah, with
evidence, please see the answers to questions no.
9104 and
12295. 

Secondly: 

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”

[al-Ahzab 33:5]. 

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)”

[al-Ahzab 33:5]. 

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. 

We have discussed the
ruling on the wife using a name other than her father’s in the answer to
questions no. 2537,
1942,
4362 and
6241. 

Conclusion: 

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.