Hmm, choices, choices.
Up until now I've always maintained a firm will that I want a return to the Super Mario 64 formula. That was one of the greats, you could mostly explore the levels in any way you wanted to, return eventually with new power ups, find secrets in the hub world and go nuts with everything the game had to offer. Partially I'd love this because it brings back the exploration aspect and being able to see a level as a huge playground. Bob-Omb Battlefield is still one of the greatest Mario opening levels I've ever played, and if nothing else I just want to be able to soar with the wing cap again in a 3D environment with proper physics; not that shitty red star that only works in the Galaxy hub world.
I still want to see a game like that. But a lot of posts here make some compelling counter-arguments to it, and if anything I'd like some compromises here and there.
But if they are levels that lead to a singular goal, they have to be big. Properly fleshed out. Allow players multiple paths or possibilities of getting across the level, be they short cuts, higher paths or otherwise.
Do not make them Galaxy styled. I'm probably one of the few people who thought this once playing it but I did not dig Galaxy much at all. Most open worlds were small even in comparison to Super Mario Sunshine, and the levels that were linear were shallow and one-note. They offered challenge but they were rather lacking in substance for me. After Super Mario 64 managed to get away with the fact that the levels were all basically boxed in where you could see past the invisible walls, Galaxy pretty much just says "fuck it" and has all the levels suspended in mid-air composed by blocks or paths forming roads. Despite the game looking beautiful I always found it lazy, and it meant that the game had to be scripted in it's design. That kind of level design was suited better for the likes of Super Mario 3D Land, which I still think has more fun level design than Galaxy does.
Setting? Hell go nuts, but make it feel like it's brand new. I'd love a world that goes the ways of Paper Mario 2, or Super Mario World. Make the player feel part of something big, don't just enclose them in an empty space. Games like Banjo-Kazooie and Banjo-Tooie does this well, it makes the levels and worlds feel big. You'd have mountains, forests or such towering over you, and large pathways to explore. I'd love that kind of feeling in a 3D Mario.
As for the soundtrack? Again, look to Paper Mario 2 as reference. Feels adventurous, catchy and exciting. Groovy and modern. Doesn't need orchestra to back things up, and no wacky "bah's" and such. A soundtrack can feel grand without them instrumental production values.
And bring back the jump dive.