Omni weapons have been mentioned, and Electic/Biotic Slash were included in one poster's breakdown so how about adding melee weapons as a thing, whether as general gear or class specific gear?
Existing melee weapons:
Shield/Omni-shield
Omni-blades (single or double)
Swords
Krogan warhammer
Omni-bow (not technically melee but whatever)
Omni-bayonettes (not actually used but they increase melee damage)
Rough idea for class distribution:
Soldier: Omni-bayonette, combat knife (monomolecular or vibroblade or whatever makes it cut real good and not seem out of place in a future battlefield)- provides a significant boost in melee damage, can be used in effective CQC
Sentinel: Omni-shield. Self explanatory
Engineer: Omni-blades- can customize and add effects to a greater extent than any other class. Single or dual.
Infiltrator: Sword- used for Shadow Strike and other stealth-type kills.
Adept: Omni-bow. Can use bolts in conjuction with biotics to explode things.
Vanguard: Scaled down hammer- Charge and upswing, nova and downswing. You're welcome.
These wouldn't have to be locked down or even used. But using an off-class melee weapon might carry some penalty (similar to using an untrained gun in ME1 perhaps) with options to offset it somewhat.
Edit: Whoops forgot the Phoenix/Dragoon energy whip things. Those could be a bonus.