What I would do:
Visuals - If 2D, give it that anime style that Wario Land: Shake It! had (but now in HD, obviously) or like the art they use for promotional Mario content in general. If 3D, maybe like the 3D World style from Super Mario Maker 2, but polished up a bit more.
Story - Now a very interesting story would be nice, but at the very least give us a little more than the NSMB games did. The original Super Mario Bros. trilogy and Super Mario World may have had simple stories compared to like say the RPGs, but there was definitely more going on in it than "save Peach, stop Bowser; go!" I'd have some cutscenes added in too. The obligatory opening and ending sequences of course, but maybe some in between each World as you progress through the game, and some short banter before boss battles. And yes, I'd finally give Mario some lines to say. They did it for Fortune Street, so why not here?
Level Design - Well, moreso themes than designs, but while I'd fine with a couple twists on the usual world themes, maybe add a few that are uncommon, like a giant city or maybe an entire haunted valley where the Castle level is instead one huge Ghost House! Oh, and a Star world with levels that actually fit the visuals of the world map; that was one of my biggest nitpicks with the New Super Mario Bros. games...!
And maybe a multiplayer mode where you race across levels you clear in the game; you could even give it online functionality!