She ran away from her family’s home and married a Christian; is she entitled to a share of her father’s estate?

Firstly: 

The factors that determine whether one person inherits from
another are three: blood ties (kinship), marriage and wala’ (relationship
between a freed slave and his former master). And the impediments to
inheritance are three: being a slave, being the murderer of the deceased,
and being of different religions. The fact that your sister ran away from
the family home does not alter the fact that she is the daughter of your
parents. This means that she comes under the heading of kinship by virtue of
which she is entitled to inherit from them so long as she has not
apostatised from Islam. 

The fact that she ran away from the family home does not
deprive her of her right to inherit from your father, even if she has stayed
away from her parents’ home for a long time. And we do not know of any
difference of scholarly opinion concerning that. 

Secondly: 

Marriage of a Muslim woman to a non-Muslim man is a major
sin, and there is scholarly consensus that it is haraam and that the
marriage contract is invalid. Allah, may He be exalted, says (interpretation
of the meaning):

“And give not (your daughters) in marriage to
Al-Mushrikoon till they believe (in Allah Alone) and verily, a believing
slave is better than a (free) Mushrik (idolater, etc.), even though he
pleases you”

[al-Baqarah 2:221] 

“O you who believe! When believing women come to you as
emigrants, examine them, Allah knows best as to their Faith, then if you
ascertain that they are true believers, send them not back to the
disbelievers, they are not lawful (wives) for the disbelievers nor are the
disbelievers lawful (husbands) for them”

[al-Mumtahanah 60:10]. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him)
was asked: A man married a Muslim woman then it became apparent that the man
was a kaafir. What is the ruling on that? 

He replied: 

If it is proven that the man mentioned was a kaafir at the
time of the marriage contract and the woman was Muslim, then the marriage
contract is invalid because it is not permissible, according to the
consensus of the Muslims, for a kaafir to marry a Muslim woman, because
Allah, may He be glorified, says (interpretation of the meaning): “And
give not (your daughters) in marriage to Al-Mushrikoon”[al-Baqarah 2:221]
and “then if you ascertain that they are true
believers, send them not back to the disbelievers, they are not lawful
(wives) for the disbelievers nor are the disbelievers lawful (husbands) for
them”

[al-Mumtahanah 60:10]. 

End quote from Fataawa Islamiyyah, 3/230 

It says in a statement of the Islamic Fiqh Council: Marriage
of a kaafir to a Muslim woman is haraam and is not permissible according to
scholarly consensus. And there is no doubt concerning that, because of what
is implied by shar‘i texts. 

End quote from Fataawa Islamiyyah, 3/231 

Any Muslim woman who is aware of that (and still does it)
deserves to incur sin and deserves to be subjected to the hadd punishment
for zina. But if a woman was unaware of this ruling, sin is waived in her
case, but it is not permissible for her to remain in that marriage, because
it was never valid in the first place. 

See also the answer to question no.
8396 

Thirdly: 

It seems from your question that your sister has not declared
any apostasy from Islam, but there remains the issue of her marriage to a
non-Muslim and her saying that her children are free to choose between Islam
and Christianity. So we say: 

1.It seems that her marriage to a
Christian man is an invalid, haraam marriage. If she did that regarding it
as permissible when she was aware of the prohibition on it, then that
constitutes apostasy from Islam, and the apostate cannot inherit from his
Muslim relative. If she did not regard it as permissible, then she is a
sinner, as mentioned above, but that does not prevent her from inheriting.

2.With regard to her saying that
her children are free to choose between Islam and Christianity, that is a
false notion and a great evil, because Islam is the true religion and
nothing other than it is valid or acceptable from anyone.

It should be noted that one of the things that invalidate
one’s being a Muslim and that constitutes apostasy is not regarding as
disbelievers the People of the Book and the mushrikeen, or having some doubt
as to whether they are disbelievers, or regarding their religion as valid.
See the answer to question no.
31807 and 6688 

What you have to do is advise your sister and explain the
seriousness of what she has said, and call her to leave her husband if he
does not become Muslim. 

To sum up the issue of her inheritance: if she remained
Muslim and did not regard as permissible her marriage that she knew was
invalid, and she did not regard the religion of the Christians as valid or
doubt that they are disbelievers, then she may inherit. Otherwise if she has
done any of these things that invalidate her being a Muslim, then you have
to advise her and ask her to repent in front of scholars in your country. If
she repents, all well and good, otherwise she is an apostate and has no
right to inherit from your father. 

And Allah knows best.