What would you want your character to be able to do? What kind of Items do you think should you be able to use/create?
Depends on if the game has a "real world" setting or if it can have monsters and such. Unreal beings would present a good additional sense of being exposed and defenseless. I'm imagining a game where you find things you weren't expecting and have to find ways to evade and run from them for the majority of the game, climbing up trees or getting in water if they can't swim. End-game items could give you a fighting chance. You can do this with normal animals too, but monsters add a lot more variety to the kind of experiences they offer.
If you're going for a realistic cast-away experience, the game could focus on getting resources and experimenting to discover "recipes" for things. Probably a mix of an xp system and actually player experimentation. Rare materials that has to be gathered in an unusal way (eg: burning a substance off the ceiling of a cave) or that only exists in dangerous locations (eg: a flower that grows in the middle of steep cliffs) would give the player interesting goals and problems to solve.
Would you like to see other intelligent life on the island, or do you want to be really alone?
I think it'd be most interesting if you felt alone for most of them game then discovered something intelligent late game, perhaps hidden in an area that's very hostile or something you meet if you follow a string of hints.
What core stats would you like to manage? (Classic examples would be hunger & thirst)
Hunger, Thirst, and Stamina come to mind immediately. I think you should model health as a wound system rather than an hp system. eg: instead of "you took 5 damage", it'd be "You received a bruise on your head" or "you broke your left arm". Dealing with specific wounds give it a much stronger survival feeling than managing a health bar. You could also have skills (things like climbing, cooking, spear throwing, ect) to manage. If you wanted to try something rarely done, you could have to manage your character's mood and do things to make yourself happy (eat good tasting food, play instruments you crafted, go fishing, ect) to preform at your best.
What do you NOT want to see in a game like this? Maybe you played a similar game and something about it really annoyed you.
I played an island survival game that had repetitive quests from the natives. If you do have quests, try to make them more unique and interesting than “Give us five banana because we're too lazy to go ourselves”.
The game should give the player the opportunity to feel like you're slowly mastering the island, but make sure the player can never feel completely safe. They should be able to make parts of the island relatively safe, but if an area is 100% safe, it can alter the tone of the game in a negative way.
Finally, make sure the player has some direction. Dropping them somewhere and saying “Have fun” usually won't work. If people want that, they'll probably go play Minecraft instead of your game.
What kind of threats for the player do you think would be possible, apart from dying of thirst and hunger?
Dangerous creatures(either animals or monsters), illness/disease, dangerous weather, quicksand/quickmud, poisonous things, pissing off natives(if you choose to have them), active volcano, risk of down sharp drops (when exploring caves or gathering resources on a cliff), disturbing the ecosystem (eg: killings too many tigers increases the pig population which means pigs will be eating plants you might want), finding a safe place to sleep, drinking contaminated water, and the risk of wound infection come to mind.
Any other suggestions for features you'd love to see?
If the island or some aspects were procedurally generated, that would do a lot to improve its longevity. Including roleplaying things that don't necessary relate to survival, like decorations for your shelter, would add a lot.
If you aren't going fully realistic, I'd love to see something mythical and mysterious that you can unfold. Something strange and difficult to study that can be uncovered. That would fit perfectly with the isolated feeling of an island survival scenario.