Becoming a Buddhist is about changing mindset and living a harmless life. It also depends on which tradition you are following. I understand that some "sects" would require the adoption of a new name. As for Theravada tradition, what one needs is to practise the Buddha's teachings to become a better and wiser person. However if you wish to have a "Buddhist" name you can adopt one. If you have contact with your teachers or monks, you can request them to choose a name for you.
Usually "Buddhist" names are in the language of the "sect". You would have Tibetan, Chinese, Japanese, Pali, and what have you. In Theravada tradition, it will be in Pali.
Hope this helps.