Is this an oceanside bear? Then I too would go with the laser sharks.
Oh, one more. Release an imprisoned dragon.
Later, he can have another adventure killing the dragon.
crack the bear’s skull with a typical stone ax up close if you need excitement or go with the some form of braced spear.
He could enlist the help of the old lady who swallowed a horse, and hope she has one good gig left in her.
Milk of poppy, mixed with honey.
Dig a deep hole. Make sure there are no tree roots in the hole. Put a [del]half-empty[/del] [del]three-quarters-empty[/del] jar with a little bit of honey smeared on the inside down at the bottom of the hole. Make sure the jar is shaped so as to allow easy ingress for a bear’s head, but not quite so easy out-gress.
The bear will smell the honey, enter the pit, put his head into the jar to lick the honey out, and not be able to remove his head. Assuming that you have dug the hole so as to avoid providing the bear with anything to break the jar against, it will eventually suffocate.
In the event that you trap an elephant instead of a bear, I suggest you just start over, get another jar, and dig another hole. Angry elephants can be dangerous.
Don’t even need to wait for the infection. When you have pooh on a long, sharp stick you already have one dead bear.
I think I am detecting a theme.
To show he’s compassionate, he wasn’t *intending *to kill the bear, just drive it off somehow. He’s forced into a situation where killing is necessary.
To show he’s brave, he could have easily escaped the bear himself, but stays to face it in order to save someone else. Perhaps a little orphan girl has wandered into its path.
To show he’s clever, he has no conventional weapons at his disposal. He must use some everyday object or objects in a MacGyver-like fashion to slay the bear. Perhaps there’s a scythe lying nearby and he cuts the bear’s throat. Or he has only his riding gear and uses a rein to strangle it. That would show he’s strong, too.
Wait a minute. It’s your damn story! Think it up yourself! 
If he’s going to be a hero, then the slow death by poison/pierced stomach doesn’t seem to do it, plus it would hard for him to claim credit.
A spear is good, unbraced better (if unrealistic - but I think it’s a kind of a king thing to do (although I may be thinking of a boar)). If he’s big and stealthy, he could perhaps sneak up on it while it was feasting on the village cow and crush its skull with an axe.
There’s a scene in Sword of the Storm (a fantasy novel by David Gimmell) where the hero (I think he’s younger than 15 at the time, maybe around 12?) kills a bear by stabbing it in the heart. (The bear also mauls the kid into a fatal coma and he’s only saved by magic).
Said kid eventually grows up to be a hero and a king and it’s one of his first steps on that road, so uh, probably don’t want to copy that.
Thanks all - some really great ideas here. Magic is out as it isn’t that kind of setting, and poison is out too as it isn’t heroic enough, plus an uneducated farm boy is unlikely to be familiar with poisons strong enough to kill a bear. I’m liking the braced spear idea though. I know it’s been done before, so I may have to combine in with something else to keep things original. Maybe the braced spear wounds or incapacitates the bear long enough for the boy to get in a good shot at it with a wood-axe.
Basically I want to get across the fact that yes, the boy is a big, strong lad, but more importantly that he has an uncommon amount of straight-up courage for someone his age. People assume he’s dumb because he’s big, but he just prefers to listen more than he talks. He’s a natural tactician - think Odysseus with muscles.
His brother, on the other hand, is an entirely different story. But that’s a tale for another time. 
A trebuchet. I leave the specific mechanism to your imagination, but you should really use a trebuchet for this. Will he lure the bear into the trebuchet, then launch it into a castle wall? Will he lure the bear into an enclosed area, at which point a trebuchet will fire a load of cannonballs at his butt? Who knows!
No royalties required, but it’d be nice to get a mention in the introduction.
Boy aims trebuchet at unsuspecting bear. Trebuchet misfires and completely misses the bear, who then charges the boy, while the shoddily built trebuchet tips over and squashes the bear.
You may as well have him fashion a crude trebuchet from some farm implements tied to the end of a rake, which then misses the bear, the head coming loose; the bear charges, then steps on the rake which smacks it in the head, Sideshow Bob style. Stunned, it falls heavily backwards onto the detached head, severing part of its spinal cord and paralyzing it from the waist down. The boy cautiously comes forward, kicks it in the crotch a few times (since the arms and head still move), shouting “That’s for the sheep you ate, ya sheep-eatin’ bastard!”, then waits for it to starve to death.
Wow. What kind of a king would be presaged by such fortuitous failure? 
There was the bear, eating garbage near the barn. The boy screwed up his courage and thrust the spear into the bear’s side.
The bear roared up and tried to give chase, but the wound had penetrated deeply into its organs. Before it could even finish crossing the barnyard, it collapsed and lay panting in a heap. The boy picked up his grandfather’s ax and cut the bear across its jugular to end its misery, spilling great pools of blood upon the damp earth.
OK a bit more seriously now…
The thing about a bear wounded to that extent is that it’s still very dangerous, but not really looking for a fight; getting it to charge you (as with the braced spear tactic) would be hard, as its instinct would be to avoid conflict and just prey on weak things in an opportune fashion. Since you’re looking to emphasize tactics over raw strength or courage, maybe something using a lamb as bait, even covering himself(!) in sheep blood and acting injured, to lure the bear into a trap of some kind - a pitfall, a falling cage, rope snare, etc.
Get a bear spear. Use it to kill someone who has money. Use the money to hire a professional bear killer. Tell the townsfolks that you used a bear spear to kill a bear, omit some details, and become a local hero.
The “budding hero kills a wild animal as a child” is pretty well trod (as above, its part of both the Davy Crockett and Daniel Boone legends). I think you should do something to turn it on its head a little anyways.
A valid point. Perhaps I should go with “budding hero kills group of feral goblins/kobolds/nasty humanoid monsters” instead…
Also, a bear spear may not be the kind of thing laying around. From the Wikipedia description, they’re bigger than a boar spear and rather unwieldy.
I knew a guy who had a bear skin in his office that he shot with a bow/arrow. He had set out a bait, and waited above the bait in a tree (for quite some time). The bear came along and he shot it from above.