Hey Jamar,
I am really sorry for the delay of this reply as it is a weekend for us here.
To change the badge name you have to use which language string is generate the badge title from backend > Easysocial > badge > edit the badge, example : COM_EASYSOCIAL_BADGES_STORY_TELLER_TITLE , this language string is stored in JoomlaFolder\administrator\language\en-GB\en-GB.com_easysocial.ini .
In my suggestion, you can using Joomla language override feature from backend > extension > language manager > override > select backend language > Click "NEW" > add the language string
By the way, may i know what do you mean by graphic detail of the badge?