We have a character (Alice) who has done something terrible to another character (Bob). Perhaps she made a mistake, had no control over her actions or it's something she did before a Heel-Face Turn. Anyway, she regrets it now. Very much.
Actually, she regrets it enough to offer Bob her life to make up for her mistake or action. We have two variants now:
She truly wants to be killed, and thinks Bob is the one who should do it.
She simply doesn't care if she's killed or not. She's ready to accept Bob's judgement on this, no matter the result.
In Mobile Suit Gundam 00, Lockon finds out that his teammate Setsuna was part of the group of terrorists that planted a bomb that killed Lockon's parents and sister - this was done while Setsuna was a brainwashed Child Soldier. Lockon pulls his gun on Setsuna, who just stands there, because he's kind of a Death Seeker out of guilt for his past actions. Eventually, Lockon pulls himself together and doesn't shoot him.
In Mobile Suit Gundam Wing, Heero Yuy felt so guilty for causing the death of Marshal Noventa that he went up to both Noventa's wife (offscreen) and granddaughter Sylvia (onscreen) and asked each of them to kill him. Neither did so, obviously, and Sylvia even called him a coward for that.
Similarly, towards the end of the series Lady Une (post merging of her "Lady" and "Saint" selves) handed Relena Darlian-Peacecraft a gun and asked her to kill her as punishment for having murdered Relena's adoptive father. What happened? Relena refuses to do it, and Une is brought to tears.
Kurt Godel in Mahou Sensei Negima! says this to Negi when he reveals that the Megalomesembrian senate (who he's apart of) was responsible for the destruction of Negi's Doomed Hometown (though there's no evidence that Godel himself was involved).
Rukia Kuchiki from Bleach deals with this a lot. Her guilt over killing her mentor, Kaien Shiba, prompts her to be a Death Seeker for much of the opening story (it's later specifically stated to be the reason she completely lost her powers in the beginning instead of only half). Throughout the entire story, because of her remorse, she's willing to allow herself to be killed by the Correction Forces, Ganju Shiba, and Kaien Shiba himself, after Demonic Possession.
Ritsuko Akagi from Neon Genesis Evangelion begs her friend Misato Katsuragi to kill her after revealing to her and Shinji the tank with the Rei clones, then destroying it and said clones. Misato refuses to do so, and Ritsuko is left a sobbing wreck on the floor..
In Battle Royale (novel and manga), after Yoshimi learns that Yoji intends to kill her, Yoshimi gives up and tells Yoji that he can shoot her. Instead, Yoji is overcome by emotion and does not shoot her, throwing away the gun in regret.
And then Mitsuko picks up the gun and shoots both of them.
Mireille Bouquet in Noir promises to kill her partner Kirika Yuumura after they discover the truth behind their connection. When they do, however, she cannot do it, despite a tearful Kirika begging for her to because said connection actually involved Kirika's murder of Mireille's family. And when Kirika was no older than six, even!.
InuYasha: After Naraku eggs Sango into trying to kill Rin to save Miroku's life, Sango gives her life to Sesshoumaru for her crime. Sesshoumaru was initially angry with her and threw Hiraikotsu at her but seems to forgive her after she gives Rin her gas mask as an apology.
In the Firefly fanfic Forward, River confronts Colonel Dannet, her former combat instructor at the Academy. He is so deeply distraught at what he and the Academy did to her and the other children there that he is willing to let her kill him in revenge. She is instead confused as to why she doesn't want to kill him, and forgives him when she realizes that it is because he thinks of her as a human and not a weapon.
Ventus in Broken Heroes invokes this his first stinkin' moment. Although Riku refuses his plead to kill him, even though Ventus destroyed his home when Vanitas took over, Sora has a much different perspective. So Ven becomes an Atoner, and is implied to be a Death Seeker. He gets that wish, and gets murdered by Saïx after being around for 18 chapters.
Rosario Vampire: Brightest Darkness: As part of his Dark and Troubled Past, Dark was forced by his summoner/former master to kill numerous snow women. When he meets Mizore's mother in Act II after saving her from execution at the hands of Fairy Tale, he promptly offers up his life to her in penance, but since he saved them, Tsurara lets him live, instead thanking him for his aid.
Serenity: After her battle programming is activated at the Maidenhead bar, River suggests to her brother that he should kill her rather than risk her hurting the crew. His response is to insist she never, ever think like that.
In Diamond Chariot, the head of a ninja clan offers Fandorin a chance to kill him to take revenge for his friends whom the ninjas slaughtered earlier. He doesn't technically have to atone for this, of course (since he is a ninja, after all), but he needs Fandorin's trust.
In the last chapter of The Neverending Storywhen Bastian puts down AURYN in front of Atreyu (though he isn't actually offering his life; "only" his sanity).
According to White Fang, this is what it means for a dog to let a human hold it.
In The Silver Chair, it's recounted that when King Caspian's son Rilian first disappeared, Caspian's friend Drinian felt personally responsible, because he'd never stopped Rilian from visiting the Lady of the Green Kirtle who eventually turned out to be a witch and enchanted and abducted him. He went straight to Caspian and confessed this, and when Caspian picked up an axe as if to kill him, he made no attempt to resist. Caspian changed his mind, dropping the axe and hugging Drinian as he said, "I have lost my Queen, and my son. Shall I lose my friend as well?"
In an episode of Stargate SG-1 Teal'C meets the son of a man he killed prior to his Heel-Face Turn. He is willing to be put on trial and executed for this murder to atone for his old crimes.
An episode of Farscape reveals that Aeryn was part of a firing squad that killed the original Pilot for Moya, a revelation which amongst other things causes the current Pilot to shut down the ship and then detach himself from it (an action that will both kill him and cause life support to malfunction, killing everyone else). Aeryn offers herself up to Pilot in exchange for not killing himself or the others; Pilot reveals that the reason why he detached himself was less about her and more about his own resurfaced guilt for agreeing to replace the previous Pilot, which arguably made her execution possible in the first place. Pilot and Aeryn then stroke each other's cheeks, showing that they have bonded again.
Ana Lucia on LOST offers this to Sayid after she accidentally kills Shannon. Sayid decides that she's Not Worth Killing.
Subverted by D'Hoffryn in Buffy the Vampire Slayer. After becoming a vengeance demon again, Anya grants a girl's wish to kill a houseful of men and then regrets it (Humanity Is Contagious caught up to her, presumably). D'Hoffryn says he will reverse the wish but it will cost the life of a vengeance demon. Anya bows her head, completely willing to give her life...and D'Hoffryn summons up Anya's friend and fellow demon Halfrek and kills her instead. "Never go for the kill when you can go for the pain", indeed.
Played straight on Angel, when Buffy and Faith meet up while the latter is attempting a Heel-Face Turn.
Buffy: If you apologize to me, I will beat you to death.
Faith: (quietly) Go ahead.
The climax of their rivalry battle in Kyoryu Sentai Zyuranger, Geki is unable to bring himself to kill his brother Burai. Burai, on the other hand, is more than willing to kill Geki, but instead Geki tells him to go ahead and kill him if it means it'll get rid of Burai's utter hatred to him. In the end, Burai can't bring himself to do it too, and then apologizes tearfully.
Once Upon a Time: At the end of "Welcome to Storybrooke", a broken Snow turns up on Regina's doorstep and begs Regina to kill her for what she did to Cora. Regina has a better idea.
In "Quite a Common Fairy" Regina herself does this. She rips out her own heart and gives it to Tinker Bell, but warns her that getting revenge leads to darkness.
Once Shiala is saved from the Thorian in Mass Effect 1, Shepard has the option of executing her, even though Shiala gives him/her help critical to his/her mission, and expresses a genuine desire to help the unfortunate colony they're on. If you choose to do so, Shiala will stoically accept your decision, kneel down, and look away as Shepard puts two gunshots in her head.
Shiala: If this is the fate you feel I deserve, then I will not resist.
Oerba Dia Vanille in Final Fantasy XIII when Sazh discovers that it is Vanille's fault that his son, Dajh, became a l'Cie in the first place and was taken by the Sanctum; and Vanille tells Sazh to shoot her because she deserves to die. Sazh feels pretty bad about aiming his gun at her, though. To the point where almost shoots himself instead.
Knights of the Old Republic: In the Light Side ending, Bastila asks the PC to kill her as punishment for her Face-Heel Turn. It's up to the player whether to do it, or try to convince her that she can redeem herself.
Snowe Vingerhut of Suikoden IV offers this to his former friend turned adversary Lazlo several times. The first few times, he's angry about his defeat and defiant; while Lazlo can instead offer to let him join the resistance movement, he rejects that. Only after he's been fully broken and realized that his good intentions didn't make him the good guy that he's fully willing to submit to Lazlo's judgment... and, potentially, to his forgiveness.
Meryl from Metal Gear Solid tells Solid Snake to shoot her, should she ever slow him down. Snake replies that he doesn't like to waste bullets.
Persona 3: Shinjiro (who is slowly dying anyway) offers to let Ken kill him in order to avenge his Accidental Murder of Ken's mother, though he does so with a warning. Ken, who has cornered Shinjiro with the intention of killing him, completely loses his nerve and can't go through with it. But then Strega shows up...
Magus of Chrono Trigger lets the party have a rematch to the death if they feel like punishing him for his misdeeds despite knowing he is but a Well-Intentioned Extremist and is not to blame for unleashing Lavos and indirectly causing Chrono's death. As he has been defeated once already, it's abundantly clear he doesn't expect to win. If you do spare him, he offers to atone for what he DID commit (such as the Mystic War in 600 AD) by becoming a Sixth Ranger in your group.
Near the end of Dragon Age II, party member Anders, a Well-Intentioned Extremist, has just blown up the Chantry with innocent clergy inside (and possibly tricked Player Character Hawke into helping). He surrenders to Hawke without a fight and says "If I pay with my life, then I pay." You are given the choice of killing him as he asks, kicking him out of the party, or keeping him around, though he will be surprised if you spare his life and fellow party member Sebastian, who was a Chantry brother, will leave in a rage.
In Kiss Of Revenge, Issei Sezaki realizes partway through his route that the protagonist is targeting him over the death of her mother twelve years before, and invites her to his apartment in order to give her the opportunity to kill him.
In Sengoku Basara 3, when Mitsunari discovers that his Number Two Yoshitsugu betrayed their ally Motochika, he claims that he's also at fault for remaining ignorant and asks Motochika to take revenge on him. Motochika refuses, telling Mitsunari that he should live on and repent.
In an arguably even more disturbing scene, it's heavily implied that Equius consciously allowed Gamzee to strangle him to death purely because, due to the trolls' caste system, he believed it was the latter's inherent right to do so.
Terra from Teen Titans commands Beast Boy to "destroy" her after realizing what she's done.
When Robin was unwillingly made Slade's apprentice behind his friends' backs, Starfire ends a standoff between her and Robin like this. She states she doesn't want to live in a world where Robin is truly evil and offers no resistance if Robin really wants to kill her.
In the first Bugs Bunny cartoon A Wild Hare, Bugs does this with Elmer but only as the set-up for one his faux death scenes.