Star Fox, I believe, is the Nintendo franchise that needs an overhaul the most. In a recent interview, Miyamoto said that he wishes the series would be more popular. The issue is simple, there not much depth to an on rail shooter. Star Fox was fun back in the SNES and N64 days because we enjoyed simpler games at that time. I really do like the Star Fox characters/universe and it has tremendous potential to be so much more, so here's what I'd like to see:
Make it an adventure game: I don't want to travel from planet to planet via an overworld map where I just click on the following planet. I want to be able to fly there, make my way there, encounter enemies, space stations on the way. Perhaps find shops where I can buy stuff to upgrade my gear/ship (more about this in my next point). I want the game to focus on story related missions in each planet. Each planet to have different civilizations to interact with. Explore the planet on foot, the land rover, hovercraft, all these different ships that we have seen in previous installments.
Customization/Upgrades: I want to be able to upgrade my arwing/any other vehicle, my outfits, gear, and weapons. I want to be able to change my arwing according to the atmosphere of the planet I am going to enter and my space suit as well.
Multiplayer: We have seen it before, and it's a ton of fun, bring it back!
There's not much more to add. The first two points really cover my concerns regarding the series.
Anything you'd like to add?