Characters in action games tend to gain access to a wide variety of weaponry. However, for the purpose of making a character more "iconic", the marketing for will often make the promotional art of their hero(es) fixate on a single weapon, which, whether or not any given player chooses to use that weapon in their playthrough, will become that character's signature weapon.
This is not necessarily the most powerful, versatile, or even the most useful weapon in a game; marketing for the Half-Life series habitually depicts Dr. Gordon Freeman with the humble crowbar in his hand, even though in-game, the crowbar is a practically useless weapon of last resort. Similarly, the Dead Space franchise tends to show protagonist Isaac Clarke carrying a plasma cutter, the first weapon available in the game, even though the plasma cutter is overshadowed by every other weapon in the arsenal (especially in Dead Space 3, in which players are invited to create their own custom weapons of unstoppable power).
Other times, a character's signature weapon will simply be the most common, in order to make it more likely that a player will actually use that weapon. Notably, art for the Halo franchise tends to depict the Master Chief carrying the ubiquitous assault rifle, which tends to be handy in all situations even if it doesn't excel in any of them... except in the case of Halo 2, which did not include the assault rifle in its arsenal, and which instead tended to depict the Chief carrying dual SMGs, in order to show off the game's new dual-wielding system.
Very occasionally, a developer will explicitly attempt to make a specific weapon a signature weapon; one of the DLC items (given away as a free pre-order bonus from specific retailers) available for Just Cause 2 is a handgun called "Rico's Signature Gun". This gun is a higher-resolution version of the handgun model from the original Just Cause. However, in that game, while the handgun was used by protagonist Rico Rodriguez, it was simply "the pistol", with no particular significance granted to it, it is only in the sequel that the weapon is treated with reverence.