Some great ideas here.
Customize character armor: As with ME3 SP, ability to piece together a suit with chosen upgrades (ie. upgraded shielding, faster shield recharge, medigel heal on kill, targeting vi, mobility, power amp, melee, grenade capacity, etc.. and all starting at level1 and futher upgrades say to level 5 maxed) but to be able to piece together an armor set relevant to the character and your playstyle. Also, light/medium/heavy armors for the different classes.
Customize character weapons: Similar to ME3 MP. Loved all the mods BW added. Perhaps even the ability to change weapon colors, add emblems, scuffs/damage.
Maps: Large, multi-piece, progressive maps similar to DAMP. This is something I would have loved for ME3MP rather than just horde mode.
Missions: Looking at this like we are agents/soliders/what have you, and these are the types of missions occuring in the background that in SP you might hear referenced in game. Anyhow, the missions are based on behinds the SP scenes, and we're playing them out. (ie what did all those 'assets' from ME3 provide towards success vs the Reapers')
Vehicles: As mentioned above, add turrets for instance. A player or two mans the turrets focused on larger units, while one player on the ground protects another player from adds, who is currently busy hacking for a few minutes. Also, ground and air vehicles, for ground/air/space combat. (BF spoiled me with vehicles!)
Community Map/Mission Creator: Also mentioned above. I love this idea. The ability to create our own maps and missions, and allow BW to choose and integrate the best of them. BW, there's so much creativity in the community, this could really expand the MP side of things.
Points/Awards: Gear the weighting more towards teamwork. Much less focus on e-peen.
Cover system: If there is to be a cover system in ME:Next, no auto-cover. Player must manually enter cover. The auto-cover thing in ME3MP got me killed too often.
Loading Cinematics: While the map is loading, some in shuttle/vehicle scene with banter between team/pilot/commander.