Nightwing
It was inspired by the story of a Kryptonian hero similar to Batman, and the more he had thought about it, the better it fit. Nightwing—dark, flying, nocturnal. The name was the perfect homage to the parents who had taught him to fly and the man who had raised him after and taught him a new way to fly, taught him to embrace the darkness so others weren't touched by it. The name held echoes of all of his childhood heroes.
Joining the force, he had found the term was also used for officers cleared for night patrol alone.
Dick: I don't need Batman, I don't want to be associated with Batman, oh, this name reminds me of Batman, I'll take it.
Seriously, he was angry and trying to pretend like Bruce didn't mean anything to him, but everybody except Bruce knew he was kidding himself. Not that I think Nightwing wasn't a healthy and necessary move for Dick at the time, I just think it's adorable.
Also, I was not able to substantiate the idea that "nightwing" is a term for police officers newly cleared to patrol alone at night, but I read it somewhere and it still makes me laugh, so I went ahead and used it.