By default, the platform does not have this option. You can use the following CSS code to hide or remove the name on the profile drop-down menu.
Please follow the steps below:
1. Go to BuddyBoss > Theme Options
2. Under Custom Codes, enable CSS
3. Append the following:
.header-aside-inner a.user-link img, .header-aside-inner a.user-link span span.user-name { display: none; }
4. Save changes