Elite Yelpers should always use their real names on Yelp, so if your name changes, you should update the name on your account as well.
When you change the name on your account, your Elite badge will be temporarily removed pending evaluation by the Yelp Elite Squad Council. Name changes are assessed on a case-by-case basis, but generally any straightforward updates that do not fundamentally change your identity - such as updating your last name when you get married or your first name when you grow out of the nickname your parents gave you as a kid - are okay. Changes that appear to confuse or hide your identity will result in the loss of your Elite status.
If you have a unique situation or any concerns about changing the name on your account, please reach out and let your Community Manager know before you submit the change.