Jason is the only one who calls Bruce Dad, and even that is rare. Damian only calls him Father and everyone else calls him Bruce. Alfred of course calls him Master Wayne. Lastly, Talia al Ghul once called Bruce Daddy, something he was completely fine with-which shocked Talia.
Dick Grayson is my favorite comics character, but I often find it hard to explain to people why he's so great without launching into decades of continuity. Since you're a much better writer than me, what makes Dick Grayson so great?
As a better writer than me put it,* he was supposed to be the next Batman, but became something better.
Not that I wouldn’t like it if he were also still Batman. He deserves that spotlight, and even wearing that costume only served to further illustrate what makes him different from Bruce: he’s basically okay. Not that he doesn’t have his own burdens, but in being taken in and being raised to not need crimefighting as a coping mechanism at best, an addiction to a cycle of violence at worst, he’s Batman’s ultimate success. He lost his world to criminals too, but he healed because instead of being shown that the world only makes sense when you force it to, he learned that no matter how bad it gets you’ll never be alone, because unlike Bruce, Batman was there for him. So yeah, he dedicated himself to becoming a detective and tactical genius and human fighting machine so that no one else will have to lose what he lost, but that’s just because he’s a really nice guy in a position to help. He smiles and bonds and teams up and hooks up and flies by the seat of his pants every once in awhile. Even with the 60s and 70s long behind us, Fun Batman lives on in him, because he’s built for that world in a way actual modern Batman isn’t allowed to be.
Of course, there’s also the great challenge of him: he’s well-adjusted. Most successful ongoing characters have some kind of fundamental flaw or conflict to drive stories (Superman’s issues with his self-image and relationships with those around him, Spider-Man’s emotional weakness, Batman’s…everything), and Dick doesn’t have that kind of issue aside from the occasional brooding session about getting out from under “the shadow of the bat”. I suspect that’s why his best stories, rather than regular adventures as Nightwing, tend to be about placing him under some harsh set of conditions running counter to his core nature (forced to act as Slades’ apprentice, becoming the new Batman, working for Spyral as Agent 37, currently taking orders from the Court of Owls). Still, just placing harsher conditions into his general setting as Nightwing and having him be forced to overcome it could yield about the same results, and once you do that, he’s skilled and versatile enough - as well as long-ranged enough as Batman’s international agent** - that you can put him into just about any kind of setting, and it can work.
* That writer being Mark Waid. I believe he’s actually gone on the record that a lot of material from his Daredevil began life as ideas for a Nightwing run that never happened.
** We all have that “damn, I thought of it first!” idea, right? Mine was Nightwing as the James Bond of the Batman family. Well, that and I drew a robot chicken in kindergarten years before Seth Green stepped in to steal my thunder.