Regardless of what they do for the reboot, there are two things that I want them to keep from the Nolan films:
1) I want the reboot to be just as dark as the Nolan films were - no campy or family friendly tone. Batman works the best when he is dark. Keep the dark tone that Nolan got right.
2) Keep the intelligence and complexity of the Nolan films. This means a story with tons of twists, tons of things that challenge you, great character development, good and strong themes, and with philoshopical/psychological arguments added in.
These two things are crucial to a Batman franchise and are part of what makes Batman who he is.
I heard some people say they can't do this because the reboot would get bashed for copying Nolan but I don't really see how the reboot keeping these 2 things means that it's copying Nolan. Nolan did not create dark tones and intelligent & complex stories with good character development. These things have been around for a long time. Heck, a lot of Batman stories in the comics have had the same dark tone as the Nolan films and the same level of intelligence and complexity along with philoshopical/psychological added in (example: The Dark Knight Returns, The Long Halloween). These are not things Nolan brought to Batman but things Nolan adapted from Batman. I also heard some people say the tone shouldn't be as dark and serious and then proceeded to say they want a more fictional/cartoony setting with costumes more like the comics. I don't see how these are related in any way. You can have a fantasy based setting with costumes like the comics and still keep the dark serious tone and the quality stories of the Nolan films. Many Batman related stories did this in the past. Best examples are Batman TAS, The Dark Knight Returns, The Long Halloween, No Man's Land, Arkham Asylum and Arkham City, etc.
For me, the best versions of the Batman universe are those that combine the realism with the fantasy aspects. Basically a world where Batman fights off mobsters, criminals, and all other crime-drama related types of people while still existing in a universe where people like Mr Freeze, Clayface, and Solomon Grundy can exist. Gotham City looks gothic but not Burton gothic, like someone else already said. It looks similar to the Gotham found in Batman TAS, the Arkham games, and The Long Halloween.
That's what I personally hope the reboot will have. It's time to combine the best elements from all Batman adaptations to make a great reboot