The theme mentioned throughout the Dark Knight Rises and Christopher Nolan's Batman films, is that Batman isn't Bruce Wayne or any single person. Batman is more than that, he's a symbol. There's a scene even when Batman mentions to detective John Blake that he should wear a mask when fighting crime.
So even if the man behind the mask were to die or go away, the legend could still live on. In the comics storyline, Knightfall, we see someone else take up Batman's mantle when Bruce Wayne's back is broken by Bane, of all people.
There's even an animated tv series, Batman Beyond, that tells a futuristic story of Batman letting someone else take up his mantle because he's too old.
So then, it's certainly possible that John Blake, as Robin, is meant to take over the role as Batman.
It's also possible that John Blake, as Robin, is meant to be his own hero.
In terms of the significance of the John Blake's real name being revealed as Robin, signifies that he was being taken in by Batman like an apprentice or sidekick at the end of the movie. In the comics, there's a whole list of characters that have been named Robin. The first Robin (Dick Grayson), eventually left Batman's side to become Nightwing.
There's also rumor that the symbol drawn throughout Gotham City has another meaning. Right side up it's the bat emblem.
Then look at Nightwing's emblem:
Now, with the chalked Bat symbol turned upside down... looks like a bird... like Nightwing's symbol!
But possibly this is reading too much into it all. Until another movie comes out, this seems to be left open-ended.