How important is it for a new Muslim to change his name?

Names carry meanings, and every person will be influenced by (the meaning of) his name, in most cases. Man is required to call himself – or his children – by good names with good meanings, so that they will be influenced positively by their names. 

Islam came to the Arabs and the non-Arabs, and it is not essential for a new Muslim to take an Arabic name, rather what is required is that he should not have a name that is ugly or carries a meaning that goes against Islam. Many Persians and Byzantines embraced Islam and kept their names, and did not change them. Indeed many of the Prophets had names that were not Arabic because they were not Arabs. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him) was asked: Does a person who becomes Muslim have to change his name like George or Joseph etc? 

He replied: 

He does not have to change his name unless it is a name that reflects servitude to someone or something other than Allaah, but it is good to change his name to something better.  So if he changes his name from a foreign name to an Islamic name, that is good, but as to whether it is obligatory, no it is not. 

But if his name was ‘Abd al-Maseeh [= “slave of the Messiah”, a name common among Arab Christians; a similar name in English cultures would be “Christopher” — Translator] or something of that nature, then he should change it, but if the name does not imply servitude to anything or anyone other than Allaah, such as George and Paul, etc., then he does not have to change it, because these names are shared by Christians and others. And Allaah is the Source of strength. 

Fataawa Islamiyyah, 4/404. 

See also the answer to Question no. 14622.