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Praise be to Allah.
All praise is due to Allah. First of all, we praise Allah the most glorified that He guided this woman to Islam and that you are now allowed to marry her under Islamic law (shari'ah). May Allah the Almighty help you to remain firm toward your faith. Out of compassion and caring for your sake, please allow me to reproach a particular aspect of your question, which is that you did not indicate any regret on your part regarding the sinful and forbidden relationship of taking a girlfriend and continuing with her in a friendship that angers and displeases Allah regardless of the extent of that friendship.Your question focused on a secondary issue regarding the ruling on changing one's name which is not as serious as what you have practiced for the past five years. Allah the most glorified has forbidden such an indecent relationship in the Qur'an (interpretation of the meaning, )
"not committing fornication, nor taking them as lovers (i.e. girlfriends)" (Al-Maa'ida, 5:5)
So, it is obligatory upon you to repent to Allah and to regret what has passed and to have the firm will not to return to the same action again, and to do good deeds and perform acts of worship to expiate the evils of that sin, and Allah is surely forgiving and merciful. (Please see also similar discussion of this issue in question #61)
As for your statement: "I deeply love my girlfriend's parents even though they rejected our religion," by Allah it is extremely astonishing how a Muslim man that believes in Allah and the last day can love people who are non-believers that don't believe in Allah and the last day and don't forbid what Allah forbids and don't practice the religion of truth and also refuse the religion of Islam??! Is this not a clear violation of Allah's statement in the Qur'an in surat al-Mujadlilah 58:22 (interpretation of the meaning):
"You will not find any people who believe in Allah and the Last Day, making friendship with and loving those who oppose Allah and His Messenger (Muhammad- Peace and Blessings of Allah be Upon Him), even if they are their fathers or their sons or their kindred"
We are not saying to you curse them or pick up arms against them; however, it is obligatory upon you to abhor them for their disbelief and their denial of the Lord of the worlds and to hate the invalid and untruthful religion that they are practicing. On the other hand, there is no objection for you to treat them with kindnesswhile disliking them in your heartand to be amicable towards them so that perhaps Allah will guide them through you. You will find no conflict or contradiction between these two ideas if you consider them carefully.
Furthermore, there is no objection to her parents living with you after marriage if they do not perform forbidden acts in the home (e.g., eating pork, drinking alcohol, etc.)
As for the question raised at the end of your letter regarding changing her name, the answer is: if the name of the person has a forbidden meaning, like the name of an idol or one of their deities or a name that means worshipping other than Allah, such as Abdul-Maseeh (worshipper of Jesus the messiah) or Abdul-Hussain (worshipper of Hussain) then it is mandatory to change it. However, if its meaning is permissible and not objectionable, e.g. if it means one of the colors such as red or green or a vocation like carpenter or baker, then it is OK and it is not obligatory to change it. However, if changing the name in official documents would cause the person much distress or harm, then it suffices to change it amongst his family and acquaintances.
Finally, I ask Allah to give you and this woman a halaal (lawful and good) and blissful life, full of imaan (faith) and taqwa (piety - God-fearing) and enriched with righteous posterity. All success is from Allah.