Ey! Yeah, the element that I hope comes back is good writing. ZING! HAHA! I'll be here all night.
I'm glad that they brought firefight back to 5. (in fact, all the recent content updates for halo 5 are really making me want to get an xbox one again if they keep it up i'll probably cave soon)
Honestly though, like dave said, I want to see the flood return. Specifically I want the weird trochaic heptameter speaking gravemind to come back.
But most of all; I want to see the sense of mystery return to the series. A big part of why I joined this community, and why I loved the earlier games was that there was a sense of mystery and wonder that pervaded everything (in fact, I saw a video recently that was a "literary analysis" of halo, and the guy claimed that halo 1 at least was more in line with being a mystery than any other genre of storytelling, its an alright vid, I can link it if people care) A lot of that has been lost in the later games. Part of that has to do with the writing; I'm not studied enough to make a serious critique on the writing, but I think that the writers of Halo in the later games (Reach and beyond, or even halo 3 and beyond, ODST excluded) have been trying to write epic space operas or sci-fi military dramas, and then trying to shoehorn in, or forgetting entirely, the element of mystery and wonder that so defined the games in their golden years.
The other part of that loss of the mystery has to do with how familiar we've become with everything in the story and the forerunner society and everything; you can only be awed and surprised by a forerunner structure so many times before it becomes fairly mundane. Especially when you aren't even trying to change our perspective on the things; like we are still just running through forerunner ruins blowing up covys and robots trying to stop them from activating and destroying everything.
But the thing is, the Halo universe is way big enough to bring that feeling back. As Dave mentioned (I think in a different thread) a political thriller, Some sort of interfaction drama between the different post-covy factions could fit the bill, especially since there are pretty much limitless possibilities for factions and groups that we haven't explored yet. Or if you want to keep riding the train of spectacle creep until the franchise completely burns; have the next set of games all about the precursor and the flood and exploring some precursor stuff thats gonna destroy the galaxy. It might be able to revive that feeling since we know next to nothing about precursors or their architechture/society and how it functions.
EDIT: added this next bit.
I guess the issue Halo has to do is balance a diffuse and thus less impactful set of characters versus a tighter group that gets more to do but can threaten to make the universe feel very small (hello, Star Wars.)
I don't think that this is really a problem that can't be overcome with just "better writing" - take Game Of Thrones for example; its got a million characters who still manage to have an impact or at least feel like they do. Granted, its a bit unfair to compare GoT with Halo
I think the solution might just be to introduce more threats in the universe; I feel like currently everything ties too neatly together. It would be nice to have multiple things that are seriously presenting a threat consistently, right now the only thing going on in the universe besides tons of little brush wars is the AI uprising (which is a big deal don't get me wrong) but it would be easier to write in more characters that also manage to get stuff done if there was more than one and a half threats facing the world. Heck, even in the early games there was always the threat of the flood/forerunner junk, and the covy war engine. Now its just the threat of cortana and the AI/forerunner, and the covy's aren't really a threat at all.
Halo's issue isn't that they have to balance more characters with less impact vs less characters with more impact, its more that they have to not constantly neuter the characters that they write; like the Arbiter, stuck in his palace fighting dumb little skirmishes unable to really do anything. Or the Infinity, which is so overpowered that if you want to write a good story you need to either disable it or not use it, which severely limits the ability to use any of the characters on board for any kind of story.
Another thing that they could do to bring back that mystery would be to just say that there is more to the halo rings than what we know. I mean, they were massive construction projects used as a last resort, its not too crazy to think they would slip as many other last resort ideas on to the rings as they could