Well I guess I could put a little stub here.
I've always imagined what adventure mode I would like the most. Competition is fun and all, but I personally get just as much or a bit more fun from the first year of single player. So here's my proposal on what could be done. I feel like the Adventure Mode from Melee was great for what it was and provided a sense of adventure and voyage with buildup to a final point - being Giga Bowser. There were even some optional fights here and there that were exciting and surprising, which is where my idea stems from!
I think the story mode should be just as long or maybe a bit more than Melee's. I mean, it shouldn't take any longer than doing a Cup or two in Mario Kart, but not too long so it doesn't feel as though it requires savepoints.
So what else could they do? I think, as many suggested, that a cool idea would be alternate paths and character encounters. Think about it, you go through adventure mode as say, Mario, and you encounter Wario picking on Luigi, or after a special event (beating the story in a certain amount of time) you encounter Super Sonic as the final boss of the story mode! If you're Bowser, you get to the Mario bros sections and you have to beat both Mario and Luigi. Even unrelated characters could be given some interaction, like if you're Pikachu and at the Mario section Peach scolds Mario for wanting to hurt Pikachu and she teams with him to take on Mario. The ideas are endless and are somewhat easy to implement.
A step further would be to have dedicated cutscenes (only as long as the Melee ones with no voices) or even just storybook clips like in Sonic and the Secret Rings. With each character going through the same adventure but with a few alterations and branching pathes would be a great way to extend the single players lifespan without it being to hard to develop. It would keep things simple like Melee but would add few custom encounters that make each playthrough a fresh experience!
A step beyond this would be having a beginning and ending cutscene. For example, Wario could be sitting at his house and watching TV, when he sees there's an "All-You-Can-Eat Buffet, FREE of charge, at Final Destination - today only." Seeing this, he rushes out of his house and starts his adventure mode. Then the ending cutscene could be him eating from a mountain of food with a KO'd Giga Bowser behind him. It's possibilities like these that make me excited!
And again, these cutscenes would be less than 10 seconds long, so it doesn't take away from the action and makes it feel like more than a swapped character in a story mode.