Starting off, I'd like to explain the visuals. I believe the look of the Ice Climbers themselves, Nana and Popo, has become an image Smash Bros players have become familiar with. I don't think this should change. I propose the characters would continue to be very cute visually while their clothing and items, as well as the world around them would be near photo realistic. Reminiscent of the visuals found in Pikmin. I imagine beautiful scenery with visuals that change through the day from astounding sunrises to terrorizing pitfalls, all on a blanket of snow and frost.

Next I picture a hub world. A small yet robust village teeming with the innocent cheeriness of Christmas. There are a few faces that you will become accustomed to and will meet plenty but it wouldn't be right to leave out normal civilians, workers, and fellow spelunkers that you could interact with voluntarily. However, I would not want any voice-overs present as it would ruin the atmosphere; with Popo and Nana themselves only speaking through actions. I imagine each person would look similar in design to Popo and Nana, with varying degrees in height and appearance like Pushmo. From there you would be able to access shops of different kinds that would allow you to purchase differing goods important to people of your profession, and a bulletin board with varying jobs and announcements, as well as a leaderboard of best times and records. Add a hot spring, home, and a few additional areas with mini-games such as ice skating and you've got a winter wonderland!

Gameplay wise, I expect something completely different. As you progress through the game I can see players encountering different mountains and paths, each with unique builds that require differing strategies to climb. As time goes by, thanks to blizzards, snow slides and the like, I would expect mountains you've already been on to change randomly with each job. By "each job" I'm implying a mission based progression system centered around the hub town and bulletin board I mention previously. Seeing as the game takes place in such a dangerous atmosphere, and Popo and Nana are considered the best regardless of their young ages, I could see plenty of people turning to them for assistance. Missions could range from finding supplies to saving lives or more.
Continuing further, I have an idea pertaining to the way they play in brawl. Seeing as these two are explorers, I can picture a new survivalist system. Making sure your body temp is kept in check and that you are constantly energized and awake is important for mountain climbers. These would need to be taken advantage of for a truly immersing experience, however, mini-games are a no-no and should be an avoided incumbent. Failing to protect your partner, or yourself, would result in you having to continue your mission alone. As for the climbing part, I suggest using something similar to Shadow of Colossus with a upgradeable grip system alongside your stamina. Now, considering that these two are quite young, I don't think a combat system would be viable. Yes, they should use their mallets but I believe that those should be limited to shattering ice, or at most stunning certain enemies. I suppose Topis (the small yeti-like enemies) would become harmless in this game and your main sources of threat would come from the polar bear and condor. I do expect more enemy types based on tundra wildlife but I don't expect much to do with fighting as this game (in my head) is a pure adventure game. I suggest wildlife be added that may be completely random, such as the Dragos from Mother 3 that could be potentially dangerous but are not inescapable as I would like the main focus to be on escaping rather than attacking, similar to the preview in Monster Hunter 4.
Next I would like for their to be a few types of multiplayer modes available. Those I had in mind were King of the Hill, Capture the Flag, Survival, Racing (climbing), and Co-Op. Any suggestions?
I would also like to address their origins. I do not believe they should be thoroughly revealed. Nintendo has set it so that the player decides what the relationship is between the duo and I believe it should stand that way, with the most we get being something as vague as "two orphans raised by the village". Other than that, I have no suggestions as that would ruin the game for me as well, haha!
Lastly, I will mention music. I picture something soothing and orchestrated, with the main theme sticking around the hub world. The main song that rings through my mind when I imagine it is "Snowman" from Mother 3.
Recap:
1)Cartoon meets life-like visuals ala Pikmin
2)Hub world featuring plenty of interactivity
3)mission based gameplay framed by mystery dungeon style stages
4)survivalist features including a grip and stamina gauge
5)no fighting mechanics, focusing on escape
6)multiplayer
7)vague back story if at all
8)orchestrated music