The Digidestined from Digimon Adventure could count as an example of this, since the summoners were "totally blind that [these guys are] going to need a lot of training to be useful."
Digimon Frontier also did this, but with a twist; they summoned a bunch of kids but only kept around the handful that actually got some results.
The protagonists of Those Who Hunt Elves were summoned before the start of the story to defeat the Big Bad, which they did immediately by blasting him with the armored tank the spell also summoned. They then spend the rest of the series gathering the parts of the spell that will let them get back to Earth.
Miaka and Yui, along with the past priestesses of Genbu and Byakko from Fushigi Yuugi.
Magic Knight Rayearth. The Princess Emeraude, the Pillar of Cephiro, summons the Magic Knights at the moment they all meet (coincidentally) in Tokyo Tower. Everyone assumes their calling is to rescue the princess from the evilZagato.
While it wasn't on purpose, The Familiar of Zero results in this. It's later revealed that he isn't exactly an "everyman".
The TV series MÄR (Märchen Awakens Romance) starts this way, though the high school kid accidentally summoned to help La Résistance of another dimension doesn't stay "normal" for long (the world of MÄR has much lower gravity than Earth, rendering him much stronger than most of the natives). (It should be noted that the series is an intentional takeoff on this kind of stories, with plenty of fairy tale references.)
The guys from the Monster Rancher anime were trying to unlock a powerful monster to help them. What they got was a kid from our world playing a Monster Rancher game beta disk.
Dog Days, because The Hero is very athletic. At least they don't summon a random person into their world.
Aura Battler Dunbines put an interesting twist on this and the Trapped in Another World tropes. The magical force called aura energy that powers Byston Well's aura battlers is actually stronger on our side, so the Big Bad's initial MO is to summon random people from the Upper Earth since they have much stronger aura potential, and the aura machine's firepower is drastically increased to WMD levels when they all cross over to our world.
Films — Animation
In The Flight of Dragons, modern-day scientist and fantasy enthusiast Peter Dickinson is summoned as a champion in a conflict between wizards (his mind gets put in a dragon body, which helps). Nobody is particularly impressed, although his summoner is prepared to trust that "antiquity" had a purpose in choosing this guy. which is proven in the end when Peter discovers that he can use his knowledge of science to cancel out any kind of magic by contemplating the logical rule it breaks, and literallyTalks The Big Bad To Death.
Justified in the book that the movie was (loosely) based on. Jim Eckert and Angie are transported into the past/alternate dimension using an astral projection machine invented by a colleague, not summoned. And Jim isn't exactly an "everyman", he's a medieval scholar, which puts him in the perfect position to understand the culture around him but work outside it using modern sensibilities and knowledge.
The Last Starfighter did a non-magic version. A con man looking for recruitment bounties wants to recruit the most talented pilot he can find in order to get paid by the Star League. He does this by seeding planets with a testing booth disguised as an arcade game called Starfighter; the one person who beats the game turns out to be a human teenager living in a desert trailerpark.
While Merlin doesn't summon the body of an everyman hero in GrailQuest, he does summon their mind. Yours.
Jon-Tom from Spellsinger by Alan Dean Foster. A Sanitation Engineer (janitor) was summoned to the world where the story takes place by a wizard searching for a more technical type of engineer. That he turned out to be a magician after all, which is what Clothahump expected an "engineer" to be, suggests it's not just Jon-Tom whose magic brings what's needed rather than what's expected.
In Prince Caspian, the Narnians were trying to summon the legendary kings and queen from the Golden Age (basically their King Arthur equivalent). They just didn't realize that the form they would come in was British schoolchildren.
The Silver Chair starts with the kids praying to go to Narnia, and besides one of them already had, so he was already 'part of the story,' and Aslan seems to prefer some form of continuity when linking up the worlds. The Last Battle doesn't involve anyone new, and Jill especially has spent the whole time since her last visit practicing skills specifically useful to a quest in Narnia. Note that thanks to Narnia Time, the entire history of Narnia occurs within the span of Diggory Kirke's life. The trips there seem to be as much for the enlightenment of English schoolchildren as to solve Narnia's problems.
William "Wiz" Zumwalt from Rick Cook's Wiz Biz series. At first (in Wizard's Bane), he appears to have no magical abilities, despite being summoned to fight against powerful evil wizards. Turned out that the summoner did pick the right guy — a computer programmer was just what was needed. Once he learns that magic can be programmed... And then improved on in the second book (The Wizardry Compiled) when to improve Wiz's original code, they bring over an entire programming team recruited at a SCA war... two of which join him in the magical world at the end of that book.
Lots of Dave Duncan's books use this premise. The Seventh Sword trilogy with Wally Smith being transported to the subject world is a good example. On Earth, Wally is a chemist who dies of encephalitis. In the other world, he is turned into a great swordsman (sort of). In this case gods needed a super-swordsman with an engineer's knowledge, since just a super-swordsman wasn't enough.
In the Spellsong Cycle by L. E. Modesitt, Jr., magic is done by a combination of song and music. The fact that music has magical effects leads to the paradoxical situation where musical theory is stunted, since the musical experimentation required to advance theory is dangerous. The summoned heroine is a professor of music whose trained singing voice, advanced knowledge of musical theory and Every(wo)man knowledge of science allows her to do things with magic that no one who born in the world could possibly match.
The first time Michael Moorcock's Eternal Champion is summoned goes like this; although it's only the mind that he's The Everyman, the body he gets stuck in is significantly more formidable. Interestingly, although recalling his Everyman hero identy staves off a Heroic BSOD near the end of that incarnation, on the instances when the Champion recalls previous incarnations he tends to adopt the identity of his first summoning, as opposed to Mr. Everyman, claiming 'it was the only time I experiened true happiness'. An odd instance of Becoming the Mask, since he's innumerable masks to choose from.
Terisa Morgan in Mordant's Need by Stephen Donaldson. What they wanted was a mighty warrior from another world armed with powerful rayguns. What they got was a lonely rich girl with microscopic self-esteem and no combat skills. They tried again. And anyone who Genre Savvy enough will not be surprised that Terisa turns out to be just the champion they needed after all. Though the guy with the rayguns helps eventually , even if he originally panics and blasts a hole in the castle.
Done in Keys to the Kingdom. To be fair about it, they were looking for someone about to die of natural causes. To be more specific, the villain, Mister Monday, wanted to avoid passing his Key onto a mortal, although he was legally supposed to do so. Instead of outright ignoring his obligation, Monday decided to circumvent it by giving the Key to a critically ill mortal who would die shortly afterwards. He could then quite legally take back and keep the Key. Unfortunately for him, he was being manipulated by his enemy, and the power of the Key saved the mortal's life. Whoops.
Discworld. Rincewind, after being thrown out of the universe at the end of his previous appearance, is summoned by Eric, who is trying to summon a demon.
The Demon Tech series by David Sherman begins with an evil wizard and a good "philosopher" each performing magic to summon assistance for his respective country. The bad guys wanted knowledge, so they wind up with a batch of 20th-century military doctrinal manuals (which they eventually manage to translate and use). The good guys wanted leadership. They get a USMC Gunnery Sergeant (in full dress uniform). His first words to them: "Who's in charge of this circle jerk?" Well, from then on, he is.
Grey Murphy, the title "Man From Mundania", comes to the land of Xanth to save the day.
Subverted heavily in Mary Gently's Grunts! Some odd piece of magic summons a U.S. Marine staff sergeant into the middle of a battle between a swarm of alien locusts and a battalion of trainee elf marines and their orc marine sergeants (yes you read that right). However, he panics at being sucked into a new dimension and flees screaming. Later on his logistics skills do help the Orcs defeat the aliens.
The War Gods has the novella Sword Brother where a Wencit, in summoning help, ends up getting a gunnery sergeant and a corporal driving a Striker.
Played with in the Magic Kingdom of Landover series where the everyman hero is summoned by the Genre Blind villain with the expectation of being useless as a hero, complete with a job interview designed to ensure uselessness. The hero is just the latest in a long line of summoned everymen who were until then as useless as expected.
In a duology within the Saga of Recluce, a spaceship suffers from a Hyperspace malfunction which lands them in a universe where magic is real. In the second book the main character ends up aiding a magic forest against The Empire, which wants to seal the forest away. The main character's companion speculates that the magic forest might have summoned them by causing the Hyperspace malfunction in the first place.
Enchantment, a lesser-known novel of Orson Scott Card: Ivan Smetski, college decathlete but otherwise normal grad student, saves a sleeping princess from a giant bear, and next thing he knows, he's in ninth-century Ukraine. Naked. It's never revealed who exactly summoned him.
The protagonist from The Sword Of Good, Hirou, is apparently snatched from our world before the story starts, and aside from a prophecy and ability to wield the titular blade, is less than stellar hero material. He doesn't take the pressure and stress well, and the casual slaughter of "evil monsters" nauseate him to the point of vomiting. The deconstructive twist ending is quite memorable.
The Ru Emerson sextet Night Threads summons 3 people- a lawyer, her sister, who is a hippie, and her teenage nephew, while they were out for a day trip in the desert.
Sergey Lukyanenko and Nick Perumov's novel Wrong Time For Dragons features a teenage girl coming to an average Russian man and asking him to come with her. He ends up in a world where magic is real and told that he is the next Dragonslayer meant to protect this world an invasion of dragons from another world. To this end, he must absorb the power of the four elemental magics. At the end, he finds out that one of the people chasing him is his own grandfather, the previous Dragonslayer, and that his grandmother was a dragon in human form, raped by his grandfather and banished to our world.
Andrei Belyanin's The Thief of Baghdad books feature an ordinary modern-day Russian man being whisked away to Arabian Nights Days by a genie at the wish of Omar Khayyám. While Omar is initially disappointed at the genie's choice (he expected a Middle Eastern man, not an infidel), Omar eventually accepts the protagonist as his student in thievery. The subsequent books have him be summoned again by the genie, although, by that point, he's hardly an "everyman" anymore.
A Mage's Power averts this trope. Eric was not summoned to the magical world of Tariatla to be a hero. Tasio regularly pulls people from other worlds and drops them there because it amuses him. Eric becomes a novice battle mage for a mercenary company during peace time.
The Beyonders has Jason and Rachel brought to Lyrian because of a prophecy that stated a "Beyonder", that is someone from another world, would defeat Emperor Maldor. Rachel turns out to be quite talented with magic but Jason spends a lot of the trilogy wondering why he was summoned.
The Castle Falkenstein backstory tells of Tom Olam, computer game designer, who is summoned into a world of Victorian Steampunk Fantasy by a mighty spell. Although he proves to be of some value, the real prize is the book he brought with him - he picked it up at a used book shop cheap, and it holds the secret to saving the day.
Traveler (not to be confused with Traveller) is based on this. The players are sucked into a fantasy/cyberpunk/whatever-genre-the-GM-is-running world with the items within a 10' radius.
Played with: the summon spell summons Miyamoto Musashi, one of history's greatest swordsmen, instead of some average high schooler with no idea what he's doing. He's summoned as a kid, yes, but still one that obviously has some combat training.
The sequel, Musashi Samurai Legend (which takes place in a different setting and different characters) also kind of fits, except that once again he's actually the hero they were trying to summon.
The Avatar/Stranger from Ultima (not to mention Lord British himself and his friends, perhaps a LITERAL Author Avatar.)
This is the basic plot for the Westwood game Nox, though it also features a twist: it is actually the villainess who summons Jack to the Land of Nox. Moreover, it happens accidentally, as she is only after the Orb that he unwittingly possesses.
Spyro 2: Ripto's Rage! has the denizens of Avalar trying to summon a mighty dragon to save their world, and get... well a mighty dragon, but much smaller and younger than they expected.
Cythera, where you're summoned from your home to a strange island by the LandKing. It's implied that the island's human population are descended from ancient Greeks swept there by a storm.
Myst has you stumbling on a mysterious book and... VOIP!
Disgaea 2: Cursed Memories starts with the protagonist's mother trying to summon Lord Zenon, the evil overlord that took over the planet, so her son could defeat him. What they got instead was Rozalin, the overlord's sheltered daughter, who they have to then protect as they take her back home. The game's huge twist is that they actually did summon the right person — Lady Zenon got tired with the constant battle of her previous existence and reincarnated herself as a baby girl.note As in, complete with a mother's womb and a blank slate. A little later, an Unknown Rival tracks her down, murders her family (and most of the village) takes over the name and effectively imprisons her while playing the role of devoted (if distant) father.
The term for this in the Touhou universe is "spiriting away," and Yukari likes to do this with mortals she finds interesting. Also, severelydeconstructed; outsiders are considered fair game for more predatory youkai unless they can find protection by nightfall.
Player characters in Wizard 101 arrive in the game world having been summoned from ours by the headmaster of the wizard school. How they promptly develop magical abilities is never really explained.
The player character of The Summoning is (unsurprisingly) recruited this way. It's not really made clear where s/he came from or what his/her life was like before, but s/he's sent into a vast dungeon full of puzzles and monsters to slay the Evil Overlord.
Captain N's spiritual successor, Alex Williams of Captain SNES.
Torg from Sluggy Freelance gets this during the "That Which Redeems" Story Arc. Justified since, while Torg is ordinary (if kinda stupid) by our world's standards, in the dimension he's summoned to, he's unique in being the only person who's not a complete, totally dedicated pacifist. So, while there are plenty of other people in that world who could fight off the Demonic Invaders, Torg's the only one actually willing to. He picks up some Badass Normal abilities along the way, too. The magic demon-killing sword powered by the blood of the innocent certainly helps.
Axe Cop features Mr. Stocker, a superhero with no powers. Immediately after being summoned to assist in a battle, he's bitten by an evil robot and turns into one himself, and is presumably destroyed. He later popped up to offer his help in fighting evil cats, only to be killed again.
Inverted in the Darkwing Duck episode "Planet of the Capes", where a planet full of superheroes has lost its only powerless ordinary guy. They fetch Darkwing Duck to replace him since, though he's a "superhero" in his own world, they know he has no actual superpowers and thus should make a perfect ordinary guy. Turns out the last ordinary guy got sick of the job and became a technology-powered Super Villain, whom Darkwing ends defeating while all the supers are useless.
In Teen Titans, this happens to Cyborg when he is pulled back in time to the Bronze Age by a witch to help save her people from monsters. Subverted twice in that Cyborg is not an "everyday" hero (he does begin to lose his powers, from lack of electricity, though) and in that the summoning was part of an evil scheme all along. It does otherwise fit the trope, however, including the part about falling in love with someone you cannot stay with.
More like Pick Up Every(wo)man Hero, but it counts: Firefly going to our world and asking Megan's help in the first My Little Pony special. Megan subsequently became a regular visitor to the fantastical world where the Little Ponies hailed from, in times of crisis and peace alike. It became a second home to her.