Nightwing 2 (lamest and most obvious, it'd just be the "New Adventures of Nightwing, or the Adventures of the New Nightwing" ugh...
Nightwing itself didn't really make too much sense, wings of the night? kinda poetic, so maybe something to go that route while associating it with the whole bat mythos, or perhaps going to the robin mythos as well.
Redwing? hahaha, sorry
Or, since this is going a new Batman Inc route...why not
Batboy or Batman? Why not? We got several Batmen now, we all know THE Batman. Or does Drake need to earn his chops more? Batboy would work, I mean, Red Robin or Robin itself isn't intimidating. But if criminals know there is a Bat-network out there and they see Batboy they will be scared if not for Batboy but the fact a BatMAN or BatMEN will be right behind,.
Make sense? Yeah? No?
Give me your thoughts, opinions, ideas, let me eat them.
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.