Wa alaykum salam wa rahmatullah,
A lot of reverts change their name, it can help them mentally transition into practising Islam and being Muslim, almost like drawing a line symbolising a new start in their Islamic life. Others feel it would be "too much" for their family to have to consider, so either they keep using their birth name, or at least not make their family use the Muslim name they have chosen.
Whatever the choice an individual makes with their names, it does not make a person any better or worse than any other Muslim.
Islam gives people the choice of keeping their names, especially surnames - these should never change. As for your first name, then as long as the meaning is not unIslamic i.e. "Christian, Christina, Christopher " etc then it is fine not to change. Islam is universal, for all people, so a lot of Muslims from different ethnic backgrounds have names which are not necessarily "Arabic" nor would they exclusively be used by Muslims. My name is Ameer, it means Leader or Prince, others have Arabic names like Rima, Jamil, yet others have names which are in Turkish or Persian, e.g. Cengir, Pinar etc Sometimes Muslims who are from Muslim families, change their names because the meaning of their name was not good. "Peer Ditta", "Ghulam Muhammad" are real life examples, meaning "Saint given" and "Slave of Muhammad" respectively.
The authentic hadith do show the preference in names like Abdullah and Abdur-Rahman. It is also good to have names after those we love such as Muhammad, A'isha, Khadija, Yusuf, Ibrahim, Omar and so on.
Ibn Umar reported: The Messenger of Allah, peace and blessings be upon him, said, “The most beloved names to Allah the Exalted are Abdullah, the servant of Allah, and Abdur Rahman, the servant of the Merciful.”
Source: Sunan Abu Dawud 4949
Grade: Sahih (authentic) according to Ibn Taymiyyah
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.