Slightly off course is Sheeta from Laputa Castle in the Sky. She's a farmgirl who is descended from the Laputian royal family, but she already knows this when we meet her (neither we nor Pazu does, though).
Jack from MÄR. Really a sidekick, but still a farmboy.
Arguably, Edward and Alphonse Elric from Fullmetal Alchemist. Their home town of Resembool is always portrayed as a farming village, although they're never shown participating. Also, their alchemy teacher Izumi Curtis is the wife of a butcher, which also lends to the rustic charm.
As for the other part of this trope, they're the sons of Hohenheim, the series' Big Good.
Hideki from Chobits moved away from his family's farm at the very start of the series to AVOID becoming this trope. He just wants to live a normal life and experience the thrills and modern marvels of the city while he pursues his own future. He certainly gets far more than what he expected, to be sure.
Don't forget Superman, who was raised as naive farmboy Clark Kent. Last son of Krypton.
Inverted with Superboy, his younger clone, who went to go live with the Kents for a "proper upbringing". Superboy has quite a bit more hubris than his progenitor, though...
Subverted somewhat by the Silver Age and Bronze Age continuity, where the Kents sold the farm before Clark started school and bought a general store in Smallville, as well as Clark becoming Superboy at age 8. Smallville's still a one-horse town, however.
Ironically, Superman's Kingdom Come rival Magog started out a farmboy too, albeit from Iowa rather than Kansas. He blew up Kansas.
Tobias Talltree of Clouded Sky is a farmboy before being chosen to become a Pokémon guide.
The Star Wars films have Luke, who was raised as a "moisture farmer" by his Uncle Owen and Aunt Beru before answering the Call to Adventure.
In an urban variant in Kung Fu Panda, Po the Panda was just a noodle vendor and cook until he was picked to be the Dragon Warrior seemingly out of the blue. Partially subverted in that his father does not object to Po accepting this destiny, just initially disappointed his son did not follow his footsteps in a calling he sincerely considers fulfilling.
The Matrix has Thomas A. Anderson starting off as a regular office worker - essentially the modern, urban version of the Farmboy - until fate said "Knock knock, Neo". The only things missing here are his hometown and parents getting destroyed - understandable as his "hometown" and "parents" are what he'll end up fighting against.
The Belgariad has (Bel)Garion the farmer's nephew. His home town was not blown up and his "aunt" Polgara turned out to be a powerful and ancient sorceress. Son of the Rivan Kings.
Durnik, the humble farm blacksmith of low birth, who insists that he's nobody important, turns out to be a signifigant part of the Prophecy and comes Back from the Dead to fulfill it, becoming an immortal sorcerer in the process and marrying Polgara. Too old to be a farm boy, but he otherwise seems to fit.
Eragon was a hunter/farmer raised by his uncle, whose home was destroyed by forces looking for the dragon egg. At one point was thought to be the Son of Morzan, a Dragon Rider who was the pupil of the Old Master Oromis before he joined Galbatorix It turns out that he was actually the son of The Obi-Wan Brom, who used to be Morzan's Lickspittle.
In King's Dragon, the first book of Kate Elliot's series Crown of Stars, Alain is the foster son of a merchant, and is raised by he and his sister, Alain's "aunt" Bel. He was not raised on a farm, but still fits the trope, because he grows up at home, doing work for his guardians and being a normal guy. Until he is visited by the Lady of Battles, taken to Count Lavastine's castle, and eventually discovers that he is his son and heir.
Subverted thoroughly by Elizabeth Moon in The Deed Of Paksenarrion: Paks is A) a girl, and B) a sheepfarmer's daughter who is... actually the daughter of the sheepfarmer and the sheepfarmer's wife too. She gets all her power through hard slogging, not having been born to it. (The first book in the trilogy is actually called Sheepfarmer's Daughter.)
In The Wheel of Time, Mat, Rand and Perrin start out the series as a farmer, shepherd and blacksmith apprentice respectively in a small rural town before answering the Call to Adventure. Lan initially calls Rand "sheepherder" frequently to lampdshade his inexperience in the ways of battle.
In The Lord of the Rings, Frodo is the equivalent of a nobleman (i.e. rich enough to live a comfortable life of leisure) but Samwise Gamgee is a gardener. Unlike most Farm Boys, though, Sam does not resist being chosen, but rather forces himself into the quest when others are chosen. Son of his Gaffer.
Richard from Terry Goodkind's Sword of Truth novels started off as a humble woodsman, before saving the Mother Confessor. He is the son of The Obi-Wan's daughter and the Big Bad, which also makes him the first War Wizard in a thousand years. He is also the first one to turn the Sword of Truth White, the Seeker of the Truth, and so on. Again, just in case you didn't know he was special. Despite all this, he insists he's a "simple woodsman."
Ged from A Wizard of Earthsea is a goatherd, son of a blacksmith, on a very rural island out on the edge of civilization — but the island is known for occasionally producing very powerful wizards.
Westley in The Princess Bride begins as a farmboy, but eventually becomes the Dread Pirate Roberts. He's not related to anyone important, though.
Subverted in the Vlad Taltos novel Athyra, which is told from the perspective of a smarter-than-average Teckla [serf] named Savn, who also has some skill with magic. From Savn's perspective, Vlad comes across as the Obi Wan figure, but the reader realizes this isn't the case, and indeed Savn does not have the happy destiny of the other characters on this page.
Subverted with Tomjon, apparent heir to Lancre and supposed son of King Verence I. Given to a troupe of actors to raise (mostly because that's how these stories are supposed to go, but also so that, if this whole king thing doesn't work out, he'll at least learn a useful trade) he was raised to be his father's son. Turns out that if you raise a king as an actor, he'll mostly turn out to be an actor.
Mort wasn't even much good at being a Farm Boy, yet a stint as the Grim Reaper-In-Training left him in a position to become a Duke. Not for very long, unfortunately.
His ancestor David is possibly the Ur Example, making this trope at least Older Than Feudalism. Played with in that instead being the last scion from a lost royal line, he is the founder of the lost royal line.
Codex Alera has Tavi of Calderon, an orphan raised by his aunt and uncle, powerless in a world where Everyone Is A Super. As he grows up, saves the world repeatedly, and generally attracts attention by being veryawesome, people begin to notice strange things about him. Such as how the mentally disabled slave owned by his uncle looks rather like legendary swordsman Araris Valerian behind the coward's brand on his face, and is uncannily good with a sword. Or how Tavi just happens to look a lot like Princeps Gaius Septimus, who was killed in battle assassinated under cover of a battle near Tavi's home shortly before he was born. Or how First Lord Gaius Sextus, Septimus's father, put Tavi through the Academy as his personal aide. Or how his bodyguard when he ends up Captain of a Legion is a brilliant, scarred swordsman named Araris... Yeah, "Tavi" is short for "Gaius Octavian."
Niko Christofolus in Paladins was merely a fisherman's son, before Nadide showed up on the scene.
"Well! - an honest and industrious farmer is one of the most useful members of society; and if I devote my talents to the cultivation of my farm, and the improvement of agriculture in general, I shall thereby benefit, not only my own immediate connections and dependants, but, in some degree, mankind at large:- hence I shall not have lived in vain." With such reflections as these I was endeavouring to console myself, as I plodded home from the fields, one cold, damp, cloudy evening towards the close of October.
Flinx from the Humanx Commonwealth novels was an urban variant, bought at a slave auction and grudgingly raised by the irascible Mother Mastiff in Drallar, the Wretched Hive capital of a rustic colony-world called Moth. He's the product of an illegal eugenics experiment that left him Cursed with AwesomePsychic Powers and the only Class A mind capable of activating a Lost Technology's galaxy-saving defense mechanism.
Bria in The Last Dove. She is raised in a rural village where she is mocked for not being able to Change. Then she finds out that she's Queen Vasi's daughter
Smallville, being the story of Clark before he became Superman, has shown him like this a lot.
Captain Kirk was born and raised on a farm in Iowa. Even though he becomes a Memetic Badass starship captain, he still identifies with Iowa as his home.
Gillian: Don't tell me; you're from outer space
Kirk: No, I'm from Iowa. I only work in outer space.
For that matter, Jean-Luc Picard grew up on his family's vinyard, although he shed himself of this trope as soon as he was old enough to enlist as a Starfleet cadet.
Not a hero, nor a boy, but the idea of the trope is invoked by Bert Cooper upon the death of the secretary Miss Blankenship:
Cooper: She was born in 1898 in a barn. She died on the 37th floor of a skyscraper. She's an astronaut.
Played with in Skins. Alo lives on a farm, and Mini's "pet name" for him is "Farm Boy."
In The Legend Of Zelda Twilight Princess, Link was a goatherd before he was pulled into a world of Triforces, shadow demons, and the neverending battle of good vs. evil. Link is possibly descended (or at least reincarnated) from the previous Links.
Other game's Links come close. Though not farmers, they often live a fairly simple life before something draws them into the adventure. Only the cartoon and direct sequels start out with a seasoned adventurer.
The main character of Baldur's Gate is an ordinary person living in a library fortress with their adoptive father. However, being the Child of Bhaal, the late god of murder, doesn't really allow a person to refuse The Call.
The Hero of Fable starts out as a farmboy, and then the whole village gets torched. The son of one of the most famous heroines around. Is also called Farmboy by the Rival and her brother.
Played with in Valkyria Chronicles in that Welkin acts like a farm-boy without actually being one. He also accepts the call without any serious objections. Of course his home-nation has a universal conscription law on its books so he's probably used to the idea of having to enlist if there's an invasion.
Dink Smallwood is a pig farmer who dreams about becoming a knight. His heritage is deliberately left wide open for the fanon to explore.
The hero/heroine of Dungeon Siege starts as you guessed it, a humble farmer.
Etrian Odyssey III has the Farmer class, which is supposed to be the weakest physically, but with plenty of skills useful for explorers to have handy.
Alex from Lunar: Silver Star for the Sega CD, and later other platforms, is a boy who lives in a small farm community. He gets to fulfill his dreams of going on adventures and becoming a Dragonmaster just like his hero, Dyne.
Evan in MapleStory. Just a normal boy living in a pig farm until he had a dream meeting a dragon. Guess what happens to him when he woke up from his dream...
In a slight subversion of this trope, in the Canon ending for Wing Commander 3, Blair goes off to be a farmboy. It doesn't take as he's recalled to active duty at the start of Wing Commander 4, now being a bitter, lonely man.
Constantly reminded of, Stahn of Tales Of Destiny from a little village in Lienea is taken for a dolt and doesn't get complimented very well by anyone except for the priestess Philia.
In Mass Effect 3, it's revealed that an agricultural unit was the first to pick up a weapon to protect it's fellow units after the Quarians turned on them. It's heavily implied this was one of Legion's memories.
The Farmer class is one option for the player to choose in Ancient Domains of Mystery. Boring, but Practical, the Farmer class can carry more weight, gather better herbs, doesn't need to eat as often, can make their own food, and levels up faster in the most balanced weapon type in the game than any other class. Who needs the innate ability to cast fireballs or pick locks on their journey to save the world when you have the ability to just survive?
Subverted with Lars from Girl Genius. According to his backstory, he was a cheesemaker's apprentice until he ran away with a circus. Later, he fell in love with Agata, but ultimately he died as an ordinary guy.
Arnie Harvey from Wisconsin, ordinary chubby kid whose family owns a dairy farm in Wisconsin. Until he gets superpowers in the Whateley Universe. He gets superstrength and superspeed. At the time of his story "Triple Threat" he is considered a campus supervillain.
Megan from the original My Little Pony cartoon. Not the daughter of anyone important (her parents don't ever come up, actually), but she definitely fits.
Speaking of ponies, two of the bearers of the Elements of Harmony grew up on farms. Applejack tried to leave as a kid but ultimately returned, feeling the family farm was her destiny. Nowadays she seems to run it, and while she has helped save the world twice shows no intention of ever giving up being a farmer. Pinkie Pie, on the other hand, happily turned away from the farm when she realised her destiny lay elsewhere, a destiny that as mentioned included saving the world twice, but mostly bringing parties and laughter to other ponies. Pinkie also claims the farm was so boring that they grew rocks and that her parents were the pony equivalent of Amish. With her being a Cloud Cuckoolander, we don't know how much of this is true.
Richard Bong, the top US Ace Pilot of all time (he got to 40 kills before being pulled from combat) was raised on a farm. He died while testing a P-80 Shooting Star on 6 August 1945. In some cases, his death shared the next day's front page with the atomic bombing of Hiroshima.
Alvin York famous American soldier in World War I, was from a farming family and was born in a two-room log cabin near Pall Mall, Tennessee.