Alright, let's see if I can come up with something notable...
Now, these comments take a lot from the Halo book so I'm not going to claim that they're totally original at all.
1. N7 Missions
I know there were technically N7 missions but let's rebrand this a little bit. Instead of the more 'firefight' modes in ME3 the N7 missions are more story-mode like than anything. You are dropped into one of maybe a dozen locales selected from the campaign (because I doubt they'd make completely new ones, but they could for DLC) with some changes to route and story and allow a group to run through it. Something like the Spartan Missions from Halo 4, though hopefully with a lot less lag. These would allow for a little bit more movement and toss in a bit of story to listen to while you are cutting your way through enemies. The Story bit will eventually get old of course but hey, it gives us the chance of being able to progress along some haunts we might remember fondly from the campaign.
2. Firefight
Sometimes, it is nice to just sit back and then suddenly go ramrod straight and swear like a sailor when a Banshee sync-kills you through a wall. I'm not saying that an instant-kill with a wallhack is nice, I'm referring more to a straight kill-or-be-killed mode. Instead of throwing in the odd objective like in ME3 multiplayer take the metagame of ME3 to the next level. Make it competitive, make it more explosive. Right now there is a little bit of competition around scores but it could be made into something a lot more expansive.
3.Don't do PVP
I'm serious. With the variety that was available in characters from ME3 a PVP scenario would be quite... Dangerous. You would have to weaken a lot of the stronger builds that have distinct advantages in order to do PVP. I mean, let's be serious an N7 Vanguard with a Geth Scanner could biotic slash you from across a map, and a Salarian Infiltrator will make you actually happy that the Shield Gate exists until he pulls out a carnifex to finish you off. If they do PVP, do it carefully and do it in a limiting factor. Would that suck? Less than if you have to try and run away from a Krogan Warlord with an adreneline package who's got a biotic hammer waiting to crush your skull in.
Those are my three for now, I'll probably think of some others. The removal of the omni-button, as other's have said, would be very-much-so appreciated of course.