You must fully upgrade any of the following weapons to Max Level: Assault Rifle, Shotgun, Sniper, and Beam.
The best weapon to do this with would be the Assault Rifle (pretty much most useful weapon in the game). To fully upgrade it will cost 6,500 credits.
Unlike the Abilities, Skills, and Melee, there is no achievement for fully upgrading all the weapons. I suggest leveling up beam a few levels as well as the Sniper and of course maxing the assault rifle. You wont really need to level beyond that, it will save you time at the end grinding for credits to level up all abilities.