Does a Christian woman have to give her child from her Muslim boyfriend a Muslim name?

Firstly: 

Your Muslim boyfriend should not have a girlfriend. If he wants to marry you, that is o.k. even if you are not Muslim, but on the condition that you are one of the People of the Book, i.e., Jewish or Christian. We ask Allaah to guide you to Islam and to help you to embrace it, and you will feel happy and it will change your life for the better, in sha Allaah (if Allaah wills). 

Secondly: 

It is not essential for your child to have an Islamic name, although it is preferable. But it should not be a name that has an ugly or offensive meaning, or a name which is distinctively the name of a kaafir (disbeliever), such as ‘Abd al-Maseeh (meaning “slave of the Messiah” – a name that is common among Arab Christians). 

For us, names have great significance. 

Ibn al-Qayyim – one of the greatest Muslim scholars – said: 

Because names carry meanings, it makes sense that there should be a connection between the name and the meaning, and that the meaning should not be like a total stranger with no connection to the name. Rather names have an impact on the things named, and the things named are affected by their names, whether they are beautiful or ugly, light or heavy, soft or rough. 

Zaad al-Ma’aad, 2/336. 

So long as your child will be a Muslim – as will you, in sha Allaah – we hope that you will not make him an object of mockery to others or make him stand out because of his having a name that is different from the names of other Muslims. 

As the child has not yet been born, you still have the opportunity to choose a good name for him, because it may be difficult to change it after he grows up. 

Thirdly: 

You could choose some good names for boys and girls from a book by Shaykh Bakr Abu Zayd entitled Tasmiyat al-Mawlood. Please also see question no. 7180. 

And Allaah knows best.