Imoen, from the Baldur's Gate games, especially the second one. Though at times very much vulnerable to Break the Cutie, a few minutes later she would be right back to her old cheerful self. "It's just like old times... well, except for the torture and all."
Originally it was planned that she would really be broken and transform into the slayer forcing the protagonist to kill her. If you have Throne of Bhaal, you can download (and install) Ascension to see how she was supposed to suffer the same fate AGAIN, which was only stopped by time issues.
The Little Sisters in BioShock are surprisingly cheerful and carefree for being slaves that are forced to extract a magical substance from corpses lying around in a destroyed city, using huge syringes. In the last level of BioShock 2, you gain temporary control of a Little Sister and see Rapture through her eyes, which looks like a bright and happy fairy tale castle. And then reality breaks in reminding you that you are in fact still in a submerged dystopia where almost everyone is completely insane and security systems specifically hate you.
The conscript of Command & Conquer Red Alert 3, basically they are the cannon fodder of the Soviet forces, and they are not even good at that since is ridiculously easy to kill them, still they will always salute you with phrases as "I make Premier proud!" and "Haven't we won yet?", now that's to have spirit.
Oh, and yeah, they are also Keets and Ditzes, I mean, you won't expect someone to say "We march to victory" when facing an entire batallion of far better enemy soldiers or "They have television in there?" when garrisoning a building in the middle of a warzone.
Corpse Party has Seiko Shinohara. Her mother disappeared a few years earlier, leaving her to take care of her siblings while her father worked, but she's still pretty cheery (though it's implied that it's something of a coping mechanism), even when she's pulled into the hellish dimension of Tenjin Elementary School. This bites her in the ass when her friend Naomi, after a close brush with a ghost, calls her creepy for being so cheerful in the midst of their terrible situation. They argue and separate. Seiko calms down pretty quickly and gets ready to send Naomi a text apologizing for their fight, only to be confronted by something. When Naomi next finds her, it's in a stall in the girl's bathroom. Hanging from a support beam. Oh, and Naomi later finds out that the person who hung Seiko in the bathroom was her being controlled, meaning the last thing Seiko saw was her best friend kick a bucket out from under her. Naomi damn near crosses the Despair Event Horizon and succumbs to The Corruption when receives a text from Seiko, the same text that Seiko had been preparing to send her before she died. The text's title? Re: No hard feelings.
Flonne from Disgaea, even after ending up as a servant to Laharl and getting involved in a whole load of Netherworldly and Celestial conspiracies never gives up her ideals of love and justice.
Her underling Artina in Disgaea 4 was like this as a human. Deconstructed as she was like this in a Crapsack World and was killed for her selfless actions.
Merrill in Dragon Age II is quite cheerful, to the point of gushing about how exciting it is that someone got mugged right outside her front door after leaving her Dalish clan on very bad terms and relocating to the Kirkwall Alienage.
A Hawke with the Sarcastic personality can come across as this, but you don't have to look hard to see they're a Sad Clown just trying not to fall apart.
Sunny from Evolve. She cheerfully interacts with every other hunter, regardless of how much they return the sentiment. She actually manages a degree of success, getting several of the more anti-social individuals to start trusting the team or at least come out of their shell. Even while in the middle of a hunt, she takes the time to talk about how cute some of the wildlife is.
Moira Brown in Fallout 3 is a perfect example of this. Even if you set off the nuclear bomb in her town, causing the radiation to turn her into a Ghoul, she will still retain her cheerful personality, realizing that her being a Ghoul means that she now has a willing test subject for the research that she had been wanting to do on Ghouls. Of course, it is possible to destroy her optimistic outlook by convincing her that her idea for a "wasteland survival guide" is misguided. (This will earn you the "Dream Crusher" perk, and make Moira better at fixing your weapons and armor, as she now focuses all her energy on running her shop.)
That said, crushing her hopes is punished by the game; the "Dream Crusher" perk is nowhere near as useful as the perks you get for actually finishing the guide. It's also incredibly hard-hitting on your Karma.
An argument could be made for Aerith from Final Fantasy VII, who was orphaned by violence, raised in a dirty, dangerous slum, ekes out a living for herself and her mother through selling flowers on the street, lost her first love in tragic circumstances (that are only probably unknown to her) and is being relentlessly watched and sometimes chased by the Yakuza-esque enforcers of a tyrannical Evil Corporation. Despite all this, she has a cheery, upbeat personality and only occasionally shows glimpses of sadness and/or oddness that are more connected to her unusual powers than her past. She's The Pollyanna, and that's probably why she had to die.
Tifa is like this too: Eternally optimistic, despite having a backstory involving getting sliced with a sword, watching her father die, having her hometown burned down, and not only having her childhood friend skip town on her a few years previously, but not even helping her after the aforementioned slicing, due to Heroic B.S.O.D.. Not to mention almost having a city dropped on her when she's an adult.
Laguna Loire of Final Fantasy VIII could be called this. He finally hits it off with his longtime crush and the very next day he's forced to jump off a cliff in order to escape a group of enemy soldiers. He's later seen in a quaint village and finds out that said crush thinks him dead, and has moved on and gotten married. Then he falls in love again and gets married, only to have his adoptive daughter get kidnapped, forcing him to search for her and leave his pregnant wife behind. He finally finds his daughter and has to send her, alone, back to his wife. Unfortunately, his wife dies in childbirth and since no one can find Laguna (who didn't even know she was pregnant), both of his children are sent to live in an orphanage and he doesn't see either of them again until they're adults. Despite this, he's as cheerful as ever years later, and rambles about needing love and friendship to complete one's mission when he isn't being totally psyched about being on a spaceship.
Selphie from the same game. Relentlessly cheerful, to the point of announcing that breaking a friend out of a government facility will be "like a picnic! We're going to have fun!" Some lines seem to show that she's scared of not being happy.
Selphie is more like the original Pollyanna in that she's more of The Woobie. Once she's in battle, that sunny attitude goes away and she does have her sad moments like when Trabia Garden is destroyed.
Tidus from Final Fantasy X is perhaps the best-known male example. Despite everything he's gone through with an abusive father, then loosing everything he has ever known, he remains positive and upbeat. A refreshing change from the usualFinal Fantasy hero.
And then there's Shelinda, who is the single most upbeat priestess in the entire Church of Yevon.
Final Fantasy XIII gives us Vanille; it becomes increasingly apparent as the plot unravels that it's a coping mechanism, although she's most certainly an honest bubbly person to some extent, as she retains the trait when her issues are settled. She breaks down twice: once when it comes out into the open that she indirectly ruined Sazh and his son's lives by keeping Fang uninformed of their Focus, and again when Fang tricks her into admitting she's faking amnesia. The second time almost gets her killed.
In Grand Theft Auto: Vice City, in the VCPR radio station, Jenny Louise Crab is a merciless deconstruction of this trope: not only she's a sickeningly cheerful Genki Girl, to the point of not even feeling bad about having her foster family brutally murdered, but the setup also strongly implies that Jenny is addicted to hard stimulants in an effort to ignore these memories.
Sora from Kingdom Hearts maintains this personality for the majority of the series. Even with brief Heroic BSODs he normally returns to this personality, mainly thanks to help from his friends.
Kairi tries to be this through most of her screentime in the first Kingdom Hearts game, but she sometimes lets her inner worries slip.
Kelly Chambers, Shepard's Yeoman in Mass Effect 2 is surprisingly upbeat and friendly. Rather than reacting to some of your most vicious party members with a lot of fear, she instead sees them with a certain degree of curiosity. In fact, when you rescue and recruit Garrus, she notes that she's compelled to hug him and tell him it'll be all right for everything he's been through.
Kelly Chamber remains upbeat and friendly even despite the likelyhood of post traumatic stressfollowing her life changing experience in the collector base but reveals that she's actually become a Stepford Smiler. In the third game, she refuses to return to the Normandy if you ask her because of the nightmares she still has.
Kasumi Goto. She is eternally optimistic, always reflecting that things could always be worse... even when trapped inside a derelict Reaper
If you go 100% Paragon and take Charm options whenever available, Shepard him/herself drips with this, especially in the first Mass Effect. Getting him/her angry is extremely difficult, diplomacy trumps violence every time, and all species are equal and valuable in his/her eyes.
One of the major songs from the MOTHER series is called ''Pollyanna'', and the lyrics are from the perspective of, well, a Pollyanna. This isn't obvious in the games (Especially the first two, which have minimal character dialogue), but there is a small theme about keeping happy no matter what happens to you, and no matter who calls you naive.
Elanee in Neverwinter Nights 2 isn't exactly upbeat, but does take an insane amount of her world being shattered and rearranged before her composure finally cracks in the slightest.
In Persona 3 and Persona 4, this is how the supporting characters see the main characters. In the latter, one character wonders out loud just how the protagonist can stay so calm in situations that push the rest of the team to intense distress... then remarks that it's the reason the protagonist is their leader in the first place.
Before them was the protagonist of Persona 2: Eternal Punishment, Maya Amano, who is an extreme Polyanna. She has a toy rabbit named Mr Bunbun which she brings out whenever she gets sad in order to force herself to be happy, and she's constantly encouraging other characters to be positive, not give up, and face their fears. She's best described as being the Team Mom for the entirity of Sumaru City.
Flora Reinhold, of the Professor Layton games, is like this — particularly in the third game, Unwound Future. She has some pretty good reasons to be The Woobie; both her parents have died, leaving her as a Lonely Rich Kid, and while she has a home with the Professor and he very evidently loves her dearly, he has an unfortunate habit (from her point of view) of trying to leave her at home for her own safety. Still, she comes across as Spoiled Sweet; she doesn't have a cruel bone in her body, she likes most everyone, and even Layton's archnemesis has an obvious soft spot for her.
Smiley in Riddle School. Despite being kidnapped twice, she still lives up to her name.
Vyse from Skies of Arcadia never seems to let the overwhelming odds of what he's getting himself into bother him. He consistently shuts down anyone who tries to bring him down, follows his own heroic code, and is very thoroughly a Magnetic Hero. When his base gets destroyed, he tells his crew that it's a new opportunity to build it up even better. Nothing gets Vyse down for long.
Farah Oersted from Tales of Eternia. She is typically very upbeat and energetic (in contrast to her childhood friends — sardonic, laid-back Reid, and uptight, pessimistic Keele). She's very fond of the phrase "No problem!".
Colette Brunel from Tales of Symphonia is also like this. She smiles and constantly assures everyone that she's fine and nothing's wrong even though she knows she's going to have to die to save the world, and stays positive even after learning that her sacrifice would have screwed over the world even more.
A lot of that optimism was a mask, though.
There's reason to regard Link in the various The Legend of Zelda games as a male Pollyanna. In each game, he's more or less pushed out of his peaceful lifestyle and into massive amounts of fighting, with little to show for it at the end in the way of reward. He gets turned into different animals, stranded in foreign countries, never gets the girl (except when he does), and once even lost his life. He's basically a plaything of the Hyrulean gods. But he's still friendly, good-natured, and never turns down a request for help.
Yuyuko Saigyouji from Touhou. She lost her parents when she was young, then later found out that she has the power to invoke death on anyone, terrifying her enough to kill herself in retaliation, reincarnated as a Ghost and eventually hatching a plan to resurrect a body that was sealing an evil tree... only to find out that the seal was her very own body, and thus had to spend her life as a ghost forever. Her reaction? Be a Cloud Cuckoo Lander, act like The Ditz, obfuscate gluttony and all around be a cheery ghost that enjoys her existence while hiding her intellect.
Played to tragic effect in Undertale. On a Kill 'em All playthrough, when you meet Papyrus, he'll approach you with friendship and mercy and try to redeem you in the hope that you can still be a good person if you only tried. Even when mortally wounded and about to die, with his last breath he will still encourage you to try and do the right thing. Popular consensus is that this is far more of a Tear Jerker than any amount of breaking him, and many players report resetting the game at this point, which is probably exactly the "right thing" he was talking about.
Captain Olimar in Pikmin manages to see the majesty in a planet that is essentially a Death World to his species. It helps that every time he goes there he is accompanied by adorable little carrot people and loots enough treasure to accumulate more wealth than some countries.