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Bows: Slingshots are the most common one, however slingsnote Which makes no sense at all, since they require considerable skill to use, are very damaging, and turn up in cultures that lack bows. and crossbowsnote Which makes some sense, since it requires less training than a bow... but they're still martial weapons. are also popular.
Guns: This is much more likely to be justified, since gun-like toys are extremely common, including Water Guns and Balloons as well as dart guns and popguns. Not to mention air rifles and BB Guns are only different in their underlying mechanism and power, and indeed are classed as actual weapons in many jurisdictions. In real life, there also exist fully-functional "youth" versions of rifles and shotguns in less-powerful calibers (.410 bore and .22 LR are among the most common), though one is very unlikely to see such guns used by kids in fiction (unless it's set at a Boy Scout camp or similar).
Clubs and Hammers: More or less the same deal as swords. Pretty much anything smaller than the grown up weapon really. Maybe a pillow or squeaky hammer if they're really young.
Staves and Polearms: A mop or any similarly shaped everyday object. Weapons that resemble tools (like scythes) can also have smaller versions of said tools.
Axe: An actual axe, but of the domestic, firewood chopping variety.
Instrument of Murder: While it might make sense to see the younger Spoony Bard training with their "weapon" at a young age, Rule of Funny demands that a simpler instrument be used (recorders and harmonicas being popular because of their being thought of as "children's instruments"). Toy versions work as well.
Good Old Fisticuffs: This tends to be harder to "kiddyfy", but adorable oversized gloves, handwraps, safety gear or a simple martial arts uniform (especially if they don't wear one or customise it when they're older) can have the same effect.
Armour: Generally this takes the form of either a costume or Improvised Armour made from scraps. Sometimes they might just wear an outfit which resembles it somewhat. Helmets tend to become a Bucket Helmet.
Magic: Expect a future magic user to use a wand (actually magic or not), or own a young stage magician's kit (or see below). Especially if they come to carry a Magic Staff. Mops and umbrellas work for the latter as well.
Superpowers: Finally, expect a child who will grow up to become a Super Hero (...or perhaps not) and has yet to manifest their powers play with or own some trinket that suggests them, such as a lighter for a pyrokinetic or "grabby thing" for telekinesis.
In Berserk, Guts used a full sized sword as a kid, as the mercs raising him didn't have any kid-sized weapons...but as an adult he also scales up his weapon and uses a BFS.
In Mahou Sensei Negima!, small wands are used for magic training and as emergency magical foci. Chamo even describes them as beginner wands.
In One Piece, Yasopp uses a rifle to snipe at his foes. His son Usopp uses a slingshot for the same purpose. Usopp seems to have no intention to graduate beyond a slingshot, however, and by the looks of things, does not need to use anything more advanced.
That's because he advances the slingshot itself instead, notably with Skypea's superpowered seashells to the point where he can hit targets a normal rifle can't reach.
"Training lightsabers" can be seen in use in Attack of the Clones. The Expanded Universe elaborates on these lightsabers: they are built using the "kiddy" Bondar crystal which produces a merely zapping blade, rather than a deadly blade produced by the lethal, adult Ilum crystal. There's also a middle-ground crystal, called an Adegan Kathracite, which produces a cutting blade, but not as powerful as the real deal; Padawans tend to use these crystals in their sabers.
In Moonrise Kingdom, much of Sam's "survival gear" consists of child-size versions of outdoors equipment, including a BB rifle.
In the extended edition of The Hobbit: An Unexpected Journey, after a long prologue sequence told by old Bilbo on the eve of his going-away birthday party (featured in The Fellowship of the Ring), he recounts his first meeting with Gandalf, as a child at one of the Old Took's parties. Kid Bilbo hits Gandalf's legs with a wooden sword, foreshadowing his acquisition of his Elvish Shortsword, Sting, when he goes on his own journey instigated by Gandalf.
In Ocarina of Time Link gets the slingshot, and uses it until he can get the bow (which he can only use as an adult). Twilight Princess takes the trope even further, with Link (only an adolescent in this game) showing off his aiming skills with a slingshot, and his sword skills with a wooden sword.
In Fallout 3, the tutorial has the player learn to shoot with a BB gun on their 10th birthday.
In Kingdom Hearts which featured child versions of several Final Fantasy characters had sword-wielding Tidus use a wooden pole, and nunchaku-wielding Selphie had a jump rope. Wakka may count too, if you consider some of his spikier Blitzballs in FFX.
Many of the Keyblade wielding protagonists used wooden toy swords before having a Keyblade, either for training or just play fighting. Terra, however, had a wooden toy keyblade.
At the end of Metal Gear Solid 4, Raiden meets his son who uses a toy sword to perform the same moves he does.
Super Mario: In Super Mario World, Bowser used his Koopa Clown Car in his showdown against Mario. In New Super Mario Bros. Wii, Bowser Jr. uses his own, miniature version of the Koopa Clown Car in his attempts to take out Mario (and Luigi, Blue Toad, and Yellow Toad).
In an episode of Disney's Herculesseries, in which Hercules initially scoffs at being forced to train with a basic training javelin rather than a more heroic weapon. But at the end of the episode, he uses a basic javelin to bring down the big bad guy (having been offered his choice of any weapon by Athena) and turns down the offer of a proper spear from an (actual) Amazonian Beauty.
In the Looney Tunes short The Old Gray Hare, there's a flashback where Baby Elmer has a pop-gun which he fires at Baby Bugs. The episode also begins with an elderly Elmer obtaining a Ray Gun.