Wait, how are we defining the word "reboot?" In terms of videogames, there are two off the top of my head: story and gameplay.
In terms of story, every Nintendo IP that Nintendo created in the 80s that they still attempt an ongoing series plot line could use a reboot. That means, Zelda and Metroid, not Mario (since the whole "Save the Princess" thing is more of an in-joke now). Kid Icarus Uprising was practically a reboot. It's almost the Metal Gear Solid of the series where it continues the story, but practically ignores everything about it except for quick and dirty plot points. It also has the benefit of being MIA for nearly 20 years which is 20 years not needing to retcon things.
Earthbound is essentially the only Nintendo franchise that intelligently continued its story and we can thank Shigesato Itoi for penning each game. Even then, the series is over, its story very neatly wrapped up. The problem with Nintendo's older franchises is that they were created when plots were basically a few screens of text. When they brought back a villain, no one really cared back then. Once technology allowed more freedom, they were tasked with coming up with explanations for things they never thought they needed before. In Zelda, I prefer the whole each-game-is-the-same-legend-retold-for-new-generations theory than what Nintendo actually came up with which is the biggest mindfuck ever. For my own sanity, I pretend there is no timeline because the plotholes are glaring and annoying when I think about them.
With Metroid, I think the plot works better if we pretend the Prime series exists within its own separate universe like Yoshio Sakamoto seems to. It also work better if Other M was ignored because Fusion is still a better Metroid 4.
In terms of gameplay, I think it's hard to change it without making the games feel completely foreign. At that point, why even reboot a series? Just create a new one. For example, a Metroid reboot where Samus collects guns and ammo instead of having an arm cannon while mowing through waves of enemies is distinctly not Metroid anymore. If Nintendo were to reboot any series, start with the core of the gameplay, be mindful of its tone, and expand it from there. Very few Nintendo IPs require a total gameplay reboot. Kid Icarus Uprising was. Outside of shooting arrows and limited flight, it has next to nothing in common with its predecessors. Nintendo's major franchises like Zelda, Metroid, Mario etc. have pretty good foundations in terms of gameplay. I wouldn't want to see everything thrown out just to feel new.
What I would like to see is Nintendo challenging their own formulas. What if a boss dropped out of the sky or emerged from the ground right in the middle of Hyrule Field while you thought you were simply headed to a new location to collect the next plot coupon? What if Zelda actually died instead of her coming back inexplicably in the 11th hour of Twilight Princess? Those kind of surprises keep players on their toes. I don't think Nintendo really needs a reboot for these things to happen, but I don't think they need to avoid one either. It depends on what they want to do with it. Nintendo has been bending the rules a lot. There are cameras and robots and trains in Hyrule. A reboot would allow Nintendo to settle on what they want a series to be.
If Nintendo were to reboot Mach Rider or Ice Climbers, there's very little there to build off of in a modern context. They would probably need a Kid Icarus Uprising style upgrade.
I still advocate survival horror Ice Climbers. And I'd turn Star Fox into a comedy... with Jason Bateman as the voice of Fox and Bryan Cranston as Andross. Hmm, I was joking when I started typing that, but now I actually don't think that's a terrible idea.