Better level design, and I think puzzles - got laughed at from some of the more hardcore Xbots on the forum for saying this... but having a few puzzles would at least break up the tedium of just run through a cooridor shooting the same enemies. Having some scripted events every now and then would really work well, make you feeling you're in a movie. Halo 2 tried doing this to start with (levels where the giant spider walker was rolling through the city), but just got more and more derivative. But please, no platforming elements!! (This is one thing that I like about the Halo games - a lack of these) And no cheap fall deaths either (the hanging gardens level of Halo 2, I'm looking at you).
I'd also prefer a tighter, shorter experience... I think it took me around 12 or so hours to beat Halo 2 - A bit longer for me because I'm not a big FPS player, so it was grappling with the controls, but once I got used to it, I breezed through most of the rest of the game, till I got to a few annoying parts towards the end where I'd get attacked by nonstop waves of Brutes, and the game would conveniently forget to give me a checkpoint between battles.
Improve the story - Halo has a big mythos to draw from, so do something instead of making nonsensical scenes with talking plants.
Weapons are fine, maybe adding one or two new ones (Flamethrower?). One of the best things about Halo is that the weapons have usually been pretty well-balanced. Maybe make dual-wielding not as awkward to initiate (I hate having to drop one of the weapons I'm holding in dual-wield mode, to switch to my alt. weapon, for instance)
Better multiplayer modes - how about the ability to choose the type of game to play in Team Training, instead of having it random all the time? And, let's have online co-op.
Also, I'd like to have some good pattern based bosses (say what you want about Metroid Prime but it had some of the best FPS bosses ever in it), and more varied enemies than the little squirts, elites, brutes, insects, the annoying Flood, and the stupid things with the beams flying around everywhere.
Make the graphics look more like the outdoor Halo 1 levels and less like the ugly, muddy purple and brown levels towards the end of H2. Fix mipmapping glitches, fix the awkward cinema glitches, and improve the NPC design - Cortana looks fine but the marines look HORRIBLE, like they have a skin disease or something.
