This is the correct answer. As Hardaway's rabbit sketch made the rounds about the studio, people began to refer to it as "Bugs's Bunny". Eventually, the possessive was dropped and the name Bugs Bunny just stuck.
'And all those exclamation marks, you notice? Five? A sure sign of someone who wears his underpants on his head.' -- from Maskerade by Terry Pratchett.
This is the correct answer. As Hardaway's rabbit sketch made the rounds about the studio, people began to refer to it as "Bugs's Bunny". Eventually, the possessive was dropped and the name Bugs Bunny just stuck.