Great question!

So let's look at needs.

Platformer? No. Super Mario, Kirby, Donkey Kong, etc. More than enough,

Action-adventure? Nintendo has arguably the two best action-adventure series in Zelda and Metroid. It just needs to actually MAKE a Metroid game.

Puzzle? Picross, Dr. Mario, Pushmo, etc. Looking good there.

RPG? Surprisingly good! Monolith Soft and Intelligent Systems are two of the best in the business. When you combine Xenoblade, Fire Emblem, Paper Mario, and Mario & Luigi, it's pretty impressive.

Tactics/Strategy? Nintendo has a homegrown RTS, plus properties with potential: Advance Wars and Codename STEAM.

Racing? Mario Kart is the premiere kart racer, but Nintendo could benefit from exploring its other IPs: F-Zero, Wave Race, Excite, and Diddy Kong Racing

Shooter? There's a rail shooter in Star Fox and a third-person shooter in Splatoon. But Nintendo would do well to subsidize a first-person shooter. Maybe something exploration focused?

Sports? The industry needs more arcade sports game. Why not resurrect Mario Sluggers or Strikers, or make a COMPLETE Mario Tennis game?

Adventure? Well, no one outside of PC/indies is really exploring this genre. Maybe fund a sequel to Zack & Wiki?

Fighting? Again, surprisingly good! Nintendo has one of the premiere fighting games in Smash Bros., plus a new hit with Pokken Tournament.

Simulation? With a steady diet of Animal Crossing and Harvest Moon, Nintendo looks to have the sim base covered.

Horror? Now that Nintendo co-owns Fatal Frame, it has a foothold in the horror genre. Maybe bring back Eternal Darkness?

So I guess I'd recommend that Nintendo dust off some unused properties like F-Zero, Eternal Darkness and Wave Race, and maybe invest in a first-person shooter. Open-world games are big in the industry today, so maybe there's potential there. Although perhaps Breath of the Wild will scratch that itch.

I want Nintendo to do a Japanese style war game. Maybe based off WWII but the locales and historical figures are altered. Not exactly a FPS, more like an up close 3rd person shooter like Metal Gear or something. Yeah, I want Nintendo's version of Metal Gear!

Nintendo needs more RPGs & 3D platformer collectathon type games (mario64, galaxy)

I want Nintendo to do their own Kingdom Hearts where it'll be an action RPG with all the Nintendo worlds together. I guess it'll be like Super Smash Bros RPG.

You may want to look I to battalion wars. It leans  more towards rts than metal gear buy still a first party Nintendo ip that's just oozing potential 

RPGs, platformers, multiplayer hits, and open world adventures.

Some specifics that fall outside the core games I want from Nintendo:
I'd love to see an RPG/Platformer similar to Super Paper Mario, except without the silly paper gimmick.

I want a fresh new local multiplayer/party game like Wii Sports or Just Dance when they first came out.

Updated fitness software, I was a little disappointed that Nintendo didn't really advance their fitness software very much after the initial launch of Wii Fit. I would love to see the introduction of weights.

More casual software, I am finding myself playing more and more iPad these days, and I think Nintendo could easily smash games like Clash of Clans if they put out their own version - Miitomo is not what I was looking for. The game should have something to keep me coming back every day, some kind of progression. I REALLY like the Nintendo coins thing though. Give me an Empire building game! Or a Mushroom Kingdom builder, something like that.

I really just want a Metroid series done in the styling of Dead Space. Horror third person shooter isolated on an alien world/ship.

Metroid Prime is an action-adventure game, though.

Mario Tennis was rushed out the door, in many ways more meager than the N64 original.

Why do cutesy characters matter?

I don't know if Resident Evil, even in its early days, achieved that level of terror. I found Silent Hill accomplished that more effectively. Anyway, Nintendo co-owns Fatal Frame, which is a slow-burn scary series. 

1. Metroid in the style of Super Metroid
2. Mario extension of the Galaxy series
3. New ideas like Splatoon

I feel like they have flirted with crossing that line once or twice with the series, I just want them to commit and go for it.