Dick Grayson in his original Nightwing costume. From Tales of the Teen Titans #59 (November 1985).
In pre-Crisis on Infinite Earths continuity, the maturing Dick Grayson grows weary of his role as Batman's young sidekick. He renames himself Nightwing, recalling his adventure in the Kryptonian city of Kandor, where he and Batman meet the local hero of the same name.
Nightwing: Secret Files & Origins #1 and Nightwing: Year One tell the full post-Crisis version of how Dick Grayson gives up his identity as Robin (having been "fired" by Batman). Uncertain what to do with his new-found independence, Dick considers giving up fighting crime to study law, but he couldn't imagine his life in any other way. Turning to someone that he knows would understand, Dick asks Superman what he should be, if not Robin. In reply, Superman tells a tale of long ago on Krypton, about a man who was cast out of his family, just like Dick. He dreamt of a world ruled by justice, and set out to protect the helpless and victimized as Nightwing. Dick then decides to honor the legendary Kryptonian by renaming himself Nightwing. This tale retroactively erases the notion that anyone else before Grayson and Bette Kane ever held the titles of Nightwing and Flamebird, except for the birds and the legendary figures named after them.
In an adventure in which all of his Titans teammates are captured by Deathstroke the Terminator, and delivered to the H.I.V.E., Dick reveals his new identity of Nightwing and helps to free them with the help of Jericho. Grayson finally moves out of the shadow of the Bat, and would lead the Titans through some hard times. He endures brainwashing at the hands of Brother Blood, his relationship with Starfire would suffer due to her marriage of state and he would be deeply affected by the fact that Batman trained a new Robin (Jason Todd) only for him to be killed at the hands of the Joker (see also: "Batman: A Death in the Family").
Bruce and Dick remain at odds with each other for some time, due to feelings of hurt and betrayal. This increases after Jason's death. While serving with the New Titans, he is searched out by a now-teenaged Tim Drake, who has only one goal on his mind: for Nightwing to reprise the role of Robin. Dick flatly refuses, as he feels he can't take a step back to a position he has outgrown. It is Dick's refusal to return to the role that starts Tim down the road toward becoming the new Robin. After weeks of persuading and proving his potential, Grayson returns to Batman to plead Tim's case, with help from Alfred. Due to their arguments and the realization that Batman needs a Robin, Tim Drake becomes the third Boy Wonder.