Fuck this is going to take some thinking...
Quake series - Railgun, one shot giblets
Doom series - Super Shotgun, love the break open double barrel
Half-Life 1 - Snarks, such an odd little creature
Half-Life 2 - Crossbow, nailing people to walls never got old. Gravity Gun in Ravenholm w/ circular saw blades was $$$ though.
Halo series - Battle Rifle, probably my favorite category of weapon in all games but Halo's (specifically the one in Halo 2 with the burst fire) is my favorite
Fallout series - Mini Nuke Launcher, there is no other choice other than maybe the one from FO4 that shoots bowling balls
Tribes series - I only ever really played the first one, but I was partial to the Disc Launcher. Heavy Mortar comes in a close second.
Unreal Tournament series - Flak Cannon
Turok 1 - Fusion Cannon
Turok 2 - Cerebral Bore
MechWarrior series - PPC. Goodbye legs.
Destiny - Icebreaker, broke the game in a good way. D2 guns suck so far.
Star Wars games - Lightsaber