Is it permissible to pray for forgiveness and mercy for a kaafir who is still alive?

Allah says: “and say: "My Lord! Bestow on them Your Mercy as they did bring me up when I was young” [al-Isra’ 17:24]. My mother does not pray, and as far as I know, the one who does not pray is beyond the pale of Islam. How can it be permissible for me to pray for mercy for her in that case?
But one of the seekers of knowledge told me that it is permissible for me to pray for her in this fashion only while she is still alive, because Allah may have mercy on her and guide her to start praying. Is this true?
Is it correct for me to offer this supplication for my father? How should I say the du‘aa’ in Arabic if I pray for him? I hope you will pray for my mother and ask Allah to guide her.

Praise be to Allaah.
Firstly:

We ask Allah, may He be exalted, to guide your mother to that which He loves and which pleases Him, and to make her one of the devout women who worship Allah, and to grant you the joy of seeing her guided and becoming righteous. And we ask Him to unite your family in obedience to Him in this world and in the highest degrees of Paradise in the Hereafter.

Secondly:

What the seeker of knowledge told you about it being permissible to pray for mercy for a non-Muslim while he is still alive is correct in the sense that Allah may bestow mercy upon him by saving him from sin and guiding him to obedience, and thus transferring him from being one of those who are subject to His wrath to being one of those upon whom He bestows His mercy.

The more correct view, concerning the correctness of which the scholars are unanimously agreed, is that supplication (du‘aa’) may be made for guidance for a kaafir who is still alive, but not for mercy or forgiveness. By doing so, one will avoid an area of scholarly differences concerning this issue.

Among the evidence that this is more correct is the report narrated by al-Tirmidhi (2739) and Abu Dawood (5038) from Abu Moosa (may Allah be pleased with him) who said: The Jews would pretend to sneeze in the presence of the Prophet (blessings and peace of Allah be upon him) hoping that he would say to them, “May Allah have mercy on you,” but he used to say: “May Allah guide you and set your affairs straight.”

See the answer to question no. 43164.

Thirdly:

There is no difference of opinion among the scholars that one cannot pray for forgiveness and mercy for a kaafir who died in a state of kufr.

Al-Nawawi (may Allah have mercy on him) said:

Offering the funeral prayer for a kaafir and praying for forgiveness for him is haraam according to the text of the Qur’aan and scholarly consensus.

Al-Majmoo‘, 5/119.

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:

Praying for forgiveness for a kaafir is not permissible according to the Qur’aan, the Sunnah and scholarly consensus.

Majmoo‘ al-Fataawa, 12/489

With regard to praying for mercy and forgiveness for a kaafir who is still alive, there are numerous comments of the scholars concerning the permissibility of doing so, but not in the sense of asking for forgiveness for his shirk and kufr if he dies believing in that, or asking for mercy for him from his Lord when he has met Him as a kaafir. Rather, such supplications are to be understood as seeking to achieve that which may be a cause of his being forgiven and shown mercy, which is that he be guided to Islam. This is one of the ways in which we may interpret the words of Ibraaheem (peace be upon him) as mentioned in the verse (interpretation of the meaning): “O my Lord! They have indeed led astray many among mankind. But whoso follows me, he verily is of me. And whoso disobeys me, – still You are indeed Oft-Forgiving, Most Merciful” [Ibraaheem 14:36], and the words of the Prophet (blessings and peace of Allah be upon him): “O Allah, forgive my people, for they do not know.” Narrated by al-Bukhaari (3290) and Muslim (1792).

Ibn al-Qayyim (may Allah have mercy on him) said:

“And whoso disobeys me, – still You are indeed Oft-Forgiving, Most Merciful” – it does not say “still You are indeed Almighty, Most Wise”, because the context speaks of seeking the kindness of Allah and of du‘aa’ (supplication) i.e., if You forgive them and have mercy on them by enabling them to recant shirk and accept Tawheed, to give up sin and turn to obedience, as it says in the hadeeth, “O Allah, forgive my people, for they do not know.”

Madaarij al-Saalikeen, 1/36, 37

Badr al-Deen al-‘Ayni (may Allah have mercy on him) said, commenting on the hadeeth, “O Allah, forgive my people, for they do not know”:

What this means is: guide them to Islam with which forgiveness becomes valid, because the sin of kufr cannot be forgiven. Or it may be that what is meant is: Forgive them if they become Muslim.

‘Umdat al-Qaari Sharh Saheeh al-Bukhaari, 23/19

al-Haafiz Ibn Hajar mentioned these two interpretations in Fath al-Baari, 11/196

To sum up:

It is better to pray that your mother be guided and that Allah open her heart to doing what He has commanded her to do. But if you pray for mercy for her, there is nothing wrong with that, with the intention that Allah may bless her with guidance to do that which earns mercy, i.e., prayer.

To learn about the best way to call one who does not pray, see the answer to question no. 47425.

And Allah knows best.