Her mother wants to give her a birthday party – what should she do?

Firstly:  

With regard to this observance which people call “birthdays” (eid milaad in Arabic), Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) issued the following fatwa: 

“Everything which is taken as an ‘eid’ (something which is celebrated regularly) and is repeated each week or each year and is not prescribed in sharee’ah, is a kind of bid’ah (reprehensible innovation). The evidence for that is the fact that the Lawgiver prescribed ‘aqeeqah for the newborn, and did not prescribe anything after that. When they adopt these observances every week or every year, it means that they are making them like the Islamic Eids, which is haram and is not permitted. There are no celebrations in Islam apart from the three prescribed Eids: ‘Eid al-Fitr, ‘Eid al-Adha, and the weekly ‘Eid’ which is Friday (Yawm al-Jumu’ah). 

This does not come under the heading of customs because it is repeated. Hence when the Prophet (peace and blessings of Allah be upon him) came to Madeenah and found that the Ansaar had two festivals which they used to celebrate, he said, ‘Allah has given you something better than these: Eid al-Adha and Eid al-Fitr.’ (Narrated by al-Nasaa’i, 1556; Abu Dawood, 1134; classed as saheeh by al-Albaani in Silsilat al-Ahaadeeth al-Saheehah, no. 124), even though this was one of their customs.” 

From Sharh Kitab al-Tawheed, 1/382; see also Question # 1027. 

Secondly: 

With regard to how you should deal with your mother, I think that you should be frank with her, and tell her that this action is not permitted by Allah, and that the religion of Islam forbids this, and that as this is the case you cannot come to this party. Tell her. “Were it not for the fact that Allah does not allow it, I would have been happy to come, and I thank you, but it is not up to me or anyone else, it is up to Allah Who issues His decree and we – the Muslims – have to submit to His will; it is not permissible for us to dispute that so long as it is the command of Allah, the All-Knowing, Most Wise.” 

Tell her all of that in the best possible manner and in the kindest way. If she is convinced and appreciates that, then praise be to Allah, otherwise try to be outside the house at the time of this party, so that no one will pressure you to join in and so that you yourself will not weaken and given in. There is no sin on you for what your mother is doing, and pleasing Allah comes before pleasing any of His creation. Be certain that if your mother objects vehemently to this matter today, perhaps Allah will make her pleased with you in the future, in sha Allah.