Her Christian mother is very ill and is unconscious; can she call her to Islam?

Firstly:

We are very happy to learn that you are on your way to
embracing Islam, for there is nothing dearer to us than to see all people
guided to the religion of Islam.

The religion of Islam is the religion of the Prophets and
Messengers, the true religion, the straight path, the clear light, that
leads to the Paradise of the Most Gracious.

It is the religion of justice, mercy, goodness,
righteousness, kindness, compassion, harmony, patience, modesty, honesty,
tolerance and courage.

It is the religion that honours women greatly. It honours a
woman as a mother who must be respected, obeyed and treated kindly, and it
regards pleasing the mother as part of pleasing Allah, may He be exalted. It
states that Paradise lies at a mother’s feet, i.e., that the closest path to
Paradise is by honouring one’s mother. It forbids disobeying and angering
one’s mother, even if it is merely by saying the mildest word of reproach.
It deems the rights of the mother to be greater than the rights of the
father, and emphasises that she is to be taken care of when she is old and
weak. All of that is stated in many texts of the Qur’an and Sunnah.

By His grace and mercy, Allah causes Islam to erase whatever
came before it of sins and acts of disobedience. If a disbeliever becomes
Muslim, Allah forgives him all the sins that he committed during the time of
disbelief, and he becomes free of sins. For more information on the beauties
of this great religion, please see the following fatwas:
219, 70042, 46505 and
128862.

All you have to do now in order to enter this great religion
is to utter the twin testimony of faith (the Shahaadatayn), according to
what you are able to do. You do not have to do pronounce the Arabic letters
perfectly, and we will write the Shahaadatayn for you in English letters.
When you have uttered it, then hasten to do ghusl (full ablution), purify
yourself and offer the obligatory prayer that is currently due.

For more information, please see the following fatwas:
13569, 20267,
20815 and 13340.

We ask Allah to help you, make things easy for you, bless you
with Islam and make you steadfast therein. We are prepared to explain to you
anything that may be unclear to you and to offer any help we can.

Secondly:

You have to call your mother to Islam and continue doing so,
even if she is sick and not fully conscious. If she responds to you and
utters the Shahaadatayn, knowing that this makes her a Muslim, then that
will benefit her before Allah, may He be exalted, and it will make her one
of the people of Islam, even if she dies immediately after that, before
offering any act of worship.

The Prophet (blessings and peace of Allah be upon him said to
his paternal uncle Abu Taalib: “O Uncle, say Laa ilaaha ill-Allah (there is
no god but Allah), a word on the basis of which I will make a plea for you
before Allah, may He be glorified and exalted.” Agreed upon.

But if she dies in her current state of disbelief after
hearing of Islam – Allah forbid – then nothing can benefit her after that,
no supplication or charity and the like, because Allah, may He be exalted,
says (interpretation of the meaning):

“Verily,
Allah forgives not that partners should be set up with him in worship, but
He forgives except that (anything else) to whom He pleases, and whoever sets
up partners with Allah in worship, he has indeed invented a tremendous sin”

[an-Nisa’ 4:48]

“It is not
(proper) for the Prophet and those who believe to ask Allah’s Forgiveness
for the Mushrikoon (polytheists, idolaters, pagans, disbelievers in the
Oneness of Allah) even though they be of kin, after it has become clear to
them that they are the dwellers of the Fire (because they died in a state of
disbelief)”

[at-Tawbah 9:113].

It was narrated that Ibn Mas‘ood (may Allah be pleased with
him) said: The Messenger of Allah (blessings and peace of Allah be upon him)
said: “Whoever dies associating anything with Allah will enter Hell.” Agreed
upon.

See also fatwa no. 43164

Thirdly:

With regard to anyone whom the call of Islam reaches in a
sound manner, proof has been established against him. But if a person dies
without the call having reached him, or with it having reached him in an
unsound manner, then the matter is to be decided by Allah, for He knows best
about His creation; your Lord will never wrong anyone, and Allah has full
knowledge of His slaves. It is proven from Abu Hurayrah, from the Messenger
of Allah (blessings and peace of Allah be upon him), that he said: “By the
One in Whose hand is the soul of Muhammad, no one of this nation – Jewish or
Christian – hears of me, then dies without believing in the message with
which I was sent, but he will be one of the inhabitants of the Fire.”
Narrated by Muslim.

See fatwa no. 1244

And Allah knows best.