All info from wikipedia and mythology books but going to go for the intersexed thing to keep it like the movie. Laufey is the Queen of Jotunheim while Farbauti was the King. Thanks for reading.
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Promises to keep
Odin gracefully landed after his return journey from Jotunheim through the bifrost. He checked the precious bundle he held in the crook of his arm. Still asleep. The small being had fallen asleep during their trek from the ruined temple, where Odin had found him, to the bifrost site. Odin had ignored the still gathered army though he spared a glance and a few nods to those that greeted him. As king he knew that his return to Asgard after the final defeat of the Jotun enemy should be a time of feasts and celebration starting from the moment he stepped foot back on Asgardian soil. However he was also acutely aware of the need for discretion and secrecy regarding this newest member of his household. He had already decided that he would love this child and would do whatever he could to protect him.
The child hadn't reacted to Odin's presence as Odin had towered over him in the temple, too weakened by his time spend alone and neglected to do anything but whimper quietly and weakly kick his limbs. The wave of pity and the deep seated need to protect the child that washed over Odin was not that surprising to him. Odin was old enough to know himself, he could be cruel and sometimes he had to put the needs of Asgard over the needs of the individual but he would never have called himself heartless. Who could see such desperate hopelessness and need in one so small and frail and then walk away from it. 'Not I' thought Odin. He reached down carefully to not startle the infant, his finger touched the cheek of the child lightly as he poured a small amount of magic into the tiny body, in the hopes of sustaining him for the journey to his new home. What he hadn't expected was the change in skin colour that occurred in the child. The pink rosiness of skin that was inherent in all Asgardians spread out from the point of contact between them, the pink overshadowed the blue and suddenly, on the temple alter lay a child that, though small even for an Asgardian birth, could well have been born nothing more or less than a normal child. He picked the child up carefully, fully realising that in Asgardian skin the cold would be keenly felt by the youngster, so Odin removed one of the furs he wore to stave off the cold and wrapped the child tightly in the warmth, protecting him from the very land that had been his home until now.
Though the child now looked like an Asgardian, Odin knew however that this child would never be the same as the other children. He thought of his son, Thor, who at the time of Odin's leaving had been just over three years old. He would be five soon, Odin shook his head at that, thinking on all the times he had missed with his son. His attention was diverted back to the small being now lying in his arms, the child had been enervated by the short burst of magic and was cooing and babbling at him softly and reaching out to grasp at his beard. Odin wondered if he was the first to show this little one any attention at all as the child stared up at him with eyes that were open and trusting. He knew frost giants could survive on the snow of their home world with ease, this must have been how the child had survived alone for so long, he had obviously not been cared for, abandoned even. He sighed, he had known the parentage of the child as soon as he had seen the markings on his body. The child wore the birth markings of the king and queen of Jotunheim, Farbauti and Laufey. He sighed again as he knew the king lay slain in the field of battle and that his wife, Laufey, was currently in the Jotun palace under guard. The queen had made no mention of a child, maybe it was assumed that the child had already perished and was an issue no longer.
This was the point that Odin came to his decision, he would see this child to Asgard and there he would become a beloved member of the house of Odin. Though in time there may be hope for peace with Jotunheim through him but there was time to think about that later, his first priority for now was the child he held in his arms. He thought of his Queen, Frigga had a large heart, she would love this child well and see him raised as their son. He smiled down at the fur wrapped child, "You need a name little one." He thought for a moment, his finger going to absent mindedly stroke the child's cheek. "You shall have a strong name. You are Loki Odinson and you are my son." and with this, he pricked the tiny finger of his new son on one of the discarded knives set on the alter he still stood next to, this caused the child to wail and flail it's small limbs. He held the child as he cut his own hand and mixed their blood. "Shhhhh now little one, it is over. We are family now" he soothed the child for a long while, he needed the child to be quiet as they went through the Asgardian war camp. Luckily the energy the child had gained from the earlier transfer of magic had begun to wear off and Odin could see that Loki would soon be deeply asleep. He wrapped the child more firmly in the furs, shushing him and soothing him as he did so. It was like this that they set off together towards the bifrost site. Odin quietly confident in the decision he had made that day and Loki tucked into the crook of his arm fussing slightly as he fought sleep. Odin thought as he walked, he promised himself that he would do all in his power to love and protect this child from everything, including from the knowledge of his heritage. He promised himself also that even if one day everything backfired and all his plans came to naught, he would never regret the decision to save this child, his son.