At least when he explained it. Nightwing was used to homage two of the early influences in his life: Batman and Superman. Nightwing was Superman's identity in Krypton when he had to take on a "Batman-style" role, so it makes sense for Dick to adopt something like that.
So what would make Robin adopt a new mantle? He'd have a little bit of Batman, but also Dick Grayson in his name. I still like the name Redwing or Batwing. Redbat just sounds weird.
So, you don't like "Red Wing"? "Redwing" actually does it better for me LOL. On another hand: "Nightwing and Redwing" could be catchy, whenever Dick stops playing Batman (I miss Nightwing!).