His mother died in kufr, can he pray for her?

Praise be to Allah. 

Allah says in His Book:

{مَا كَانَ لِلنَّبِيِّ وَالَّذِينَ آمَنُوا أَن يَسْتَغْفِرُوا لِلْمُشْرِكِينَ وَلَوْ كَانُوا أُولِي قُرْبَىٰ مِن بَعْدِ مَا تَبَيَّنَ لَهُمْ أَنَّهُمْ أَصْحَابُ الْجَحِيمِ}

'Translation' {It is not (proper) for the Prophet and those who believe to ask Allah's forgiveness for the Mushrikoon, 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)} [al-Tawbah:113]

Al-Qurtubi said in al-Ahkaam [8/173]: "This aayah means that one should cut off ties of close friendship with the kuffaar, whether they are alive or dead. Allah did not allow the believers to pray for forgiveness for the mushrikeen. Seeking forgiveness for a mushrik is something that is not permitted."

Muslim narrated in his Saheeh [916] from Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (peace and blessings of Allah be upon him) said: «I asked my Lord for permission to pray for forgiveness for my mother, and He did not permit me. I asked Him for permission to visit her grave, and He granted me permission».

This hadith makes it clear that Allah did not permit His Prophet, who is the noblest of His creation, to seek forgiveness for his mother after she had died. This is what Allah forbids in the aayah quoted above, and the hadith states this clearly. Al-Shawkaani said in Fath al-Qadeer [2/410]: "This aayah implies that there are no ties of close friendship with the kuffaar, and that it is haraam to pray for forgiveness for them, or to make du'aa' for kaafirs in ways that are not permitted".

 If someone were to say that Ibraaheem prayed for his kaafir father, as he said:

{قَالَ سَلَامٌ عَلَيْكَ ۖ سَأَسْتَغْفِرُ لَكَ رَبِّي ۖ إِنَّهُ كَانَ بِي حَفِيًّا}

'Translation' {Peace be on you! I will ask forgiveness of my Lord for you. Verily, He is unto me Ever Most Gracious} [Maryam:47].

the response to that is what Allah said in the aayah:

{وَمَا كَانَ اسْتِغْفَارُ إِبْرَاهِيمَ لِأَبِيهِ إِلَّا عَن مَّوْعِدَةٍ وَعَدَهَا إِيَّاهُ فَلَمَّا تَبَيَّنَ لَهُ أَنَّهُ عَدُوٌّ لِّلَّـهِ تَبَرَّأَ مِنْهُ ۚإِنَّ إِبْرَاهِيمَ لَأَوَّاهٌ حَلِيمٌ}

'Translation' {And Ibraaheem's (Abraham) invoking (of Allah) for his father's forgiveness was only because of a promise he [Ibraaheem] had made to him (his father). But when it became clear to him that he (his father) is an enemy of Allah, he dissociated himself from him. Verily, Ibraaheem was Awwah (one who invokes Allah with humility, glorifies Him and remembers Him much) and was forbearing} [al-Tawbah:114].

Ibn 'Abbaas (may Allah be pleased with him) said: "When he died, it became clear to him that he was an enemy of Allah" (Tafseer Ibn Katheer).

And Allah knows best.

Sheikh Muhammed Salih Al-Munajjid.