This is a video game gameplay trope where a weapon in the game can be used for something other than fighting enemies; typically by bypassing obstacles or being used to solve puzzles. This could be as simple as explosives (dynamite's a popular one) which can destroy specific barriers other weapons can't touch, however on the other end of the spectrum, Metroidvania style games often give the player equipment which allows them to bypass a given obstacle but also comes in handy against the various enemies the game throws at them (for example, a jump attack which allows them to both get over large gaps and to attack flying mooks). Another popular variant is to have the player's melee weapon be an Improvised Weapon which can also be used as originally intended (e.g. repairing things, cutting through barriers, digging etc).
The reason for this trope is often rooted in the constraints of a Game Engine. If the engine was originally designed for a purely combat based game and the developer wanted to make it more puzzle based (as was the case in the 90's, when the First-Person Shooter genre began experimenting with more story based games as well as RPG Elements) then they have to modify it. Since the original engine probably already allows the player to carry and use weapons, it makes sense (from both a technical and gameplay point of view) to create weapons which can be used in puzzle solving. Another good reason is that doesn't force the player to fumble around switching from "puzzle solving equipment" to "weapons" (thus averting a Scrappy Mechanic). The Action/Adventure (especially the aforementioned Metroidvania games) genre also favours this trope, since it allows the player to both progress through the game and be more capable of fighting enemies at the same time.
Note that a weapon being used to smash through barriers (or innocent furnishings) or do anything else which any weapon could achieve doesn't count; the utility effect must be unique to that weapon (or a couple of specific weapons). Nor do weapons which happen to be the weakness of particular enemies; those fall under Achilles' Heel. Supertrope to Muzzle Flashlight.
Compare Sword of Plot Advancement for when a weapon has a non-combat use in the plot rather than gameplay. Unintended examples of this trope (for example using the recoil on a weapon to jump higher) may lead to Sequence Breaking. Within the story of the game itself, the weapon might be an Improvised Weapon (if it's a tool which could normally be used to bypass the puzzles which the game allows the player to use as a weapon) or an example of Mundane Utility (if it's something made as a weapon which just happens to be useful for solving the game's puzzles). Also related to the Inverse Law of Utility and Lethality (where abilities and weapons are more useful when they're less effective at dealing out damage). This may also function as one of the Standard FPS Guns. Often leads to a form of Equipment-Based Progression.
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The Metroid games run on this trope. The basic "Power Beam" doesn't do much outside of combat, however all the later weapons in every game can be used to remove certain obstacles and open colour coded doors.
Since every dungeon in The Legend of Zelda series contains one new piece of equipment (which will be needed to kill the boss and progress further) you can bet your last rupee that it'll have a use in both puzzle solving and combat (unless you end up against a Puzzle Boss, of course).
The LEGO series of games is full of these, practically every weapon in LEGO Indiana Jones is useful for something else (the wrench for repairs, the shovel for hidden items).
In Ōkami, Amaterasu's brush is used both as a weapon as well as pretty much everything else.
As with the Okami example, the brush in Epic Mickey serves as both a weapon and a tool (this time because of the ink based environment).
Cave Story's machine gun can be used to jump with the recoil.
Your pistol's bullets could not only kill enemy soldiers but shoot open locked doors (very useful if you don't have any keys) and speed up the unlocking of chests (better hope the chest isn't full of explosives like bullets or grenades).
Grenades can be used to destroy dangerous enemies such as SS troops. They can also be used to blow open locked doors and destroy interior walls and chests to make it easier to get around.
In the World of Mana series, many weapons can interact with the environment: Swords cut bushes, Axes smash rocks, Whips can be used to swing across gaps, and so on.
In Beyond Good & Evil, Jade's gyrodisk launcher can be used both to attack enemies and to activate switches from a distance.
In Banjo-Tooie, Clockwork Kazooie eggs function primarily as remote-controlled Action Bombs, but they also can help collect items in hard-to-reach places.
In Tomb Raider (2013), several of Lara's weapons allow her to traverse the environment in new ways. For example the axe is upgraded so it can be used to pry open obstacles, and is later replaced with the climbing axe, the shotgun can be used to blast certain types of barricade out of the way and the bow eventually gets "rope arrows".
Evolva: The first five weapons are also used to cross several obstacles. The Claws can be used to break rock walls, the Flames to burn plant walls, and the Spore to break giant boulders blocking paths. The Spikes can be used to drop explosive spores which are hanging on walls, and the Stealth technique to cross plant doors that close when approached.
First Person Shooter
In Team Fortress 2, the Engineer's wrench melee weapon also repairs/upgrades his machines. In addition, several unlockable weapons can have other utility effects, such as;
The Pyro's "Homewrecker" allows them to remove Spy saps from the Engineer's buildings. The later-released Neon Annihilator has this ability as well, though in a weakened form.
The Scout's Force-A-Nature has massive recoil, allowing what amounts to a second Double Jump on top of the one he gets for being the Scout. The Atomizer adds an actual third jump (at the cost of 10 damage), and the two can be combined for maximum jumping power.
The Medic's Crusader's Crossbow primary weapon heals allies hit with.
The Soldier's Disciplinary Action melee can be used to hit friendly players to make them and the Soldier move faster.
The Eureka Effect wrench will teleport you back to spawn when you right click with it out.
The Engineer's Rescue Ranger shotgun can be used to pick up his buildings from a distance when you right-click, at the cost of 130 metal. The Rescue Ranger's shots also repair friendly buildings they hit.
The Unreal Tournament series has the Translocator, which shoots a projectile and allows to the user to teleport to it. Aside from tactical movement utility, you can Tele-Frag enemies.
In Glider PRO, rubber bands can be used to trip switches; they're more useful for this purpose than for killing enemies, which give out nothing and respawn quickly.
Ratchet's wrench in Ratchet & Clank is a decent melee weapon, but is also used to operate a few mechanisms that resemble large bolts.
Duke Nukem II's flamethrower can be used to boost yourself up, like a jetpack.
The Apple ][ game Aztec (1982). The dynamite you found could be used to blow up enemies. It could also be used to blow holes in the floor, which could not only prevent enemies from reaching you but could also allow you to drop down to lower levels.
Mega Man 7 has several. The Freeze Cracker can be used to freeze instant-kill molten metal, Thunder Bolt can power up machines, while Scorch Wheel can burn off foliage and light up candles.
Mega Man 8: Tornado Hold can be used to rapidly damage an enemy, as well as lift Mega Man to a higher platform. Thunder Claw can be used to grab onto bars.
Mega Man 9: Concrete Shot can be used to create platforms and block lasers. The Tornado Blow allows Mega Man to do a higher jump. The Hornet Chaser can pick up otherwise-unreachable items for Mega Man.
Rocks N Diamonds has a downloadable Zelda Level which has bombs destroy certain walls and objects
Role Playing Game
In Skyrim, Mining picks (and other tools) are needed to carry out certain actions, but can also be equipped as weapons (indeed, some NPCs will use them as an Improvised Weapon).
In fact, mining ore can be done faster by wielding the pick as a weapon and "attacking" the ore-vein (just don't hit any nearby NPCs). This can also be used to mine an ore-vein an NPC is interacting with instead of waiting for them to leave. The woodcutter's ax does not function the same way unfortunately.
In Pokémon, there are Hidden Machines, which not only teach mons attacks you can use in battles, but also outside of battles to bypass barriers. In addition, some non-HM moves like Flash or Headbutt can also be used outside battle (but aren't typically needed to progress through the story.
In Arcanum, axes and hammers can be used to open stubborn doors and chests. You could also use a sword to smash them open, but it'd be more likely to break.
In Final Fantasy Mystic Quest, every weapon you get can be used in a non-combat situation: Axes obviously cut down trees, bombs open sealed passages, the claw weapons allow you to scale walls (with the ultimate version, the Dragon Claw, allowing you to grapple distant objects), and your sword can poke switches in hard-to-reach places (which only even comes up in two dungeons).
Pointy or sharp metallic weapons can be used to engrave words into the floor, though this is slow and dulls the weapon. But an athame (a specific type of dagger) can not only engrave without becoming dull, but also engraves much faster than other weapons.
A bullwhip can be used to pick an up off the ground if you're floating or riding an animal, to fish on item out of a pool of water, or can wrap around a large object and be used to pull you out of a pit.
All of Silent Hill: Homecoming's melee weapons. The knife can tear through layers of cloth/tissue, the pipe can force locks open, the axe can cut down planks boarding doors shut, etc.
In Destroy All Humans!, the Anal Probe can (if used on a human when fully charged) cause the target's brain to pop out, allowing you to extract the DNA and use it to buy upgrades.
In the Rune Factory series, you can use swords in place of your sickle and your war hammers in place of your normal hammer (though not nearly as efficiently as the normal tools). This comes in handy in RF3, where using battle hammers during mining can yield metal ores like iron and copper more often than your leveled-up tool hammer (which tends to cough up jewels and crystals).
Non-Video Game Examples
While Real Life tends to be devoid of (deliberately designed) puzzles like the ones you'd find in games, certain objects which might be used as weapons (knives, small axes, entrenching tools) tend to be useful for more peaceful purposes.
World War Z gives us the Lobotomizer, an axe that's also a shovel. So you can fortify your position and then kill the shit out of zombies. And then give them a proper burial because you're classy like that.
Any gun Smith gets his hands on in Shoot 'em Up, via trick shots. Mainly used to traverse or manipulate the environment around him during a fire fight.