Hmm neat thread...
To be honest it's quite hard for me to single out some, there are so many characters even npcs that I would like to know more about, this is usually because Zelda tends not to go into huge depth and it usually leaves more questions than it answers. So for me there are a quite a lot of characters I'd like to get to know in more depth and sometimes just brief in game information would be helpful to give us get more of an idea who exactly that characters is.
Fierce Deity
Probably one of my favourite "characters" in the series, of course he isn't really a character as such but I think the main reason I'm intrigued by him is that there is a huge sense of mystery and almost unlimited possibilities, speculation wise. The world of Termina is generally a twisted abnormality to what we know in the series it gives a sense of mystery to even the most basic npc, the lore is generally alien to that of Hyrule and I think that's what makes FD that all more intriguing.
With a this fresh world and lore that we generally no nothing about it gives us more freedom to speculate and if Nintendo were to go deeper into FD's character I think it could add something to the overall lore. The Fierce Diety himself is very similar to are young hero, Link and I think this is what interests me most, there could be some kind of connection between the two and we could be delving into the origin of Link himself. SPOILERS We have this rule of three in Hyrule mythology, basically this balance of power by three and we have the Triforce which is evenly split into three parts and given to their rightful holders. We have Ganon(dorf), Princess Zelda and Link, Ganon(dorf) origins begin at Demise while Zelda ordinates from the Goddess Hylia. Link though is just this young hero with literally any background, I suppose this is one of the many possibilities that FD could actually be.
Tingle
A lot of people see Tingle as this novelty character, kind of like the comic relief of the series, people like to make jokes about him and why not, he is an odd character that differs from the average cliché characters the series is full of. In a way though aside from being this novelty character who makes the odd appearance here and there, Tingle could in fact be much more and I know I would like to see more from him.
It's obvious that he is quite delusional, I mean he sees himself as a reincarnated fairy and who knows maybe he could be but I think that something must have happened to Tingle to actually make him what he is. If we look at his..... rather strange attire we see a tight green body suit with red underpants on the outside (yes I am describing Tingle's clothing ;p). Anyway his choice of clothing seems to be very reminiscent of what a super hero would wear, like Tingle is trying to emulate a super hero of sorts and this could be a specific Link, due to the green clothing and reference to him being a fairy in his past life/form.
Tingle also seems to don a clock necklace, maybe a reference to the whole time theme in Majora's Mask ut it always interests me when I see it. There is also the fact that Tingle seems to have this obsession with rupees and he seems to go to out of his way to go an d retrieve maps and whatnot in order to sell them to Link within said game. I feel that Tingle is a middle aged man trying to replicate something he couldn't do in the past and I would love for a Zelda game to delve deeper into his character.
Link
Now a lot of you are probably thinking "what is this guy on about lol" however besides us controlling Link do we really actually know him? We rarely get an insight into his personality, his motives, his beliefs, his background and so on. In my opinion it bothers me that the main protagonist is one of the less detailed characters. There is one main reason for this though and that's the fact that Link is a silent hero, now I know this is the case for all characters (apart from the occasional gibberish/mumbles) but Link actually never speak in the Hyrule world. The communication we see is through dialogue and Link never takes part in conversations apart from indirect communication and player choice (yes/no etc.)
I'm not proposing Link should speak or have voice acting but the fact is that Nintendo could do with giving Link a distinct personality and background. We have seen some small step ups recently, Wind Waker Link generally showed over expressive emotions through facial detail and this was a huge step up but we could generally see more interactions between characters and Link. I would also like to see Link's Legend passed down through history to give us a sense that what where doing in games is being recognised through time and this could build up this aura and heritage surrounding him.
One other thing that I would like to see regrading Link is some sort of background. We usually start the game and never delve into his past experiences, it's usually all about the present when it comes to Link. It's like the future hero is insignificant until we take control of him, this for me damages are connection and how we can relate to Link and it also damages his possible character development. Link is essentially a different perosn in each game but this gives no reason as to why Nintendo can't touch on his origins... Maybe an insight why he is where he is, why his personality is like that and why we never see him with a large family etc.