While Loki may hate his adopted father Odin for favoring Thor as his biological son, it turns out the villain actually used powerful magic to make sure he'd be adopted into Asgard's royal family in the first place. In the MCU, Odin's adoption of Loki was partly tactical, and the elder god admits that he'd hoped to one day unite the Frost Giants with the gods of Asgard. While Odin is similarly calculating in the comics, it was eventually revealed that he adopted Loki because the God of Mischief went back in time to engineer their first meeting from start to finish.
The story plays out in J. Michael Straczynski and Olivier Coipel's Thor #12. In this story, Loki visits Hela, who - on the promise of a favor in the future - lends him the power to pierce the veil of ages, travel back in time and do "that which has already been done."
With Hela's help, Loki regains his male form after some time as 'Lady Loki' and is transported back to Jotunheim, the realm of the frost giants. First, Loki ambushes Odin's father Bor, killing him and instilling a deep guilt in Odin when he predictably chooses not to try and resurrect his tyrannical father, but rather to become the new ruler of Asgard. Posing as Bor's ghost, Loki torments Odin, demanding that he repay his betrayal by one day adopting the child of one of his victims. Odin agrees, and Loki moves into his endgame.
Loki tracks down his child self in Jotunheim, offering him the power of a god in trade for engineering the deaths of his hated mother and father. Following his older self's instructions, the younger Loki goads his father Laufey into an ill-advised attack on Odin, in which the frost giant ruler is badly injured. The Loki from the future instructs his younger self to cry bitterly over his father's body, showing that he understands loyalty and family, and then to pick up his sword and rush Odin, since, "Odin respects only strength." It's this show of the attributes Odin admires that persuades him to adopt Loki in fulfillment of his promise to "Bor." As the young frost giant is carried away to become part of Odin's household, the older Loki steps in unseen to finish off Laufey, finally getting his revenge for a childhood of cruelty and neglect.
It's a chilling adventure, and one that shows Loki at his most malevolent, playing the most important people in his life off against each other and causing death and anguish in the process. Once his mission is complete, the older Loki admits he'll always wonder how events should have played out without his interference, but that he'll never regret doing what he needed to in order to achieve the station of a god. Loki becomes the author of his own success and his own suffering - a fact that doesn't counteract any of the rage he feels towards his biological and adopted families.
This time-traveling mission of manipulation fulfills Loki's desperate need to control his own destiny, but also suits his later claim to be the God of Stories, weaving the narratives that form his own past, present, and future. It turns out that while he might have been a son to both Laufey and Odin, ultimately Loki created himself - the only origin story he could ever bear to be true.