Nightwing, but without any former "Robin" persona. Batman's supposed to stalk the night and intimidate criminals, so a companion of his should also be intimidating/scary, not his little superbuddy. And he should really only train Grayson as a teenager and not bring him into fights until he's a young adult, because anything else in real world terms would make Batman a complete psychopath. Besides, it would take a long time to properly train him when Bruce also has to fight crime and play the rich playboy.
I favor Grayson over Barbara Gordon because there's an established reason why he would turn to crimefighting, instead of "because she's an accomplished, capable woman and no man's going to tell her what she can't do." Look, don't be stupid, your parents weren't murdered. Go to college or something. Maybe there could be a gender-bent Robin/Nightwing character whose real name is actually Robin, though.