Is it permissible to offer our condolences to non-Muslims when their family member passes away?
Allah SWT states:
لَّا يَنْهَاكُمُ اللَّـهُ عَنِ الَّذِينَ لَمْ يُقَاتِلُوكُمْ فِي الدِّينِ وَلَمْ يُخْرِجُوكُم مِّن دِيَارِكُمْ أَن تَبَرُّوهُمْ وَتُقْسِطُوا إِلَيْهِمْ ۚ إِنَّ اللَّـهَ يُحِبُّ الْمُقْسِطِينَ
“Allah does not forbid you from those who do not fight you because of religion and do not expel you from your homes – from being righteous toward them and acting justly toward them. Indeed, Allah loves those who act justly.” 
Imam al-Nawawi stated, “It is permissible for a Muslim to offer condolences to a non-Muslim (zimmi) and the same applies for a Muslim who has family members from among the non-Muslims…”  There are several statements or condolences remarks that a person can say to non-Muslims. Some of them are:
أَعْظَمَ اللَّهُ أَجْرَكَ، وَأَحْسَنَ عَزَاكَ، وَأَخْلَفَ عَلَيْكَ
“May Allah reward you, strengthen your perseverance and compensate over your loss” 
أَعْظَمَ اللَّهُ أَجْرَكَ، وَأَخْلَفَ عَلَيْكَ، أَوْ أَلْهَمَكَ الصَّبْرَ، أَوْ جَبَرَ مُصِيبَتَكَ
“May Allah reward you, compensate your loss, or grant you patience or cover the disaster upon you.”
أَخْلَفَ اللَّهُ عَلَيْكَ، وَلَا نَقَصَ عَدَدُكَ
“May Allah recompensate you with something better and your numbers never decrease.” 
Other types of appropriate condolences are permissible as long as it isn’t a supplication for forgiveness for the deceased. This is clearly prohibited as stated in al-Quran where Allah SWT states:
مَا كَانَ لِلنَّبِيِّ وَٱلَّذِينَ ءَامَنُوٓاْ أَن يَسۡتَغۡفِرُواْ لِلۡمُشۡرِكِينَ وَلَوۡ كَانُوٓاْ أُوْلِي قُرۡبَىٰ مِنۢ بَعۡدِ مَا تَبَيَّنَ لَهُمۡ أَنَّهُمۡ أَصۡحَٰبُ ٱلۡجَحِيمِ
“It is not for the Prophet and those who have believed to ask forgiveness for the polytheists, even if they were relatives after it has become clear to them that they are companions of Hellfire.” 
To conclude, it is permissible to offer our condolences to non-Muslim family members or those who aren’t enemies of Islam with appropriate condolences that doesn’t contradict the syarak. Hopefully, it will be a good dakwah effort to showcase the beauty of Islam. Wallahu a’lam.