In Norse legends, Tyr grew jealous of Thor, and he thought of ways to prove he was the better of the two. When the gods tried to trick the demon-wolf/god Fenrir, who was supposed to bring about Ragnarok which the gods wanted to prevent, into being leashed by the harness called Gleipnir, Fenrir would not allow it to be placed around his neck unless it could be broken. To prove this, Fenrir asked one of the gods to place their hands in his mouth as a gesture of trust; if the leash was unbreakable, Fenrir would bite off that gods' hand as a punishment. Since Gleipnir was unbreakable, Fenrir bit off Tyr's hand. You can say Tyr sacrificed a part of himself for the greater good of the community in order to prove himself superior to Thor .

When the Tuatha de Nanan king Nuadu lost his hand (some variants of the legend say his entire sword arm was severed at the shoulder) he was no longer fit to be king. The smith Credne made him a new arm of silver, but apparently, having all of your natural limbs was one of the qualifications to be king, so the title passed to Bres.

I think there can be a parallel to the Irish legend, since the implication is that Rand can no longer act as the king of the lands he has conquered because he is not longer physically fit/whole. Perhaps this is how/why Logain takes over. You could also say there are parallels to the Norse legend in that Rand proved he was better than Logain because he lost his hand to save another person (Min), whereas our impression of Logain is that of an arrogant man who allowed others to die in pursuit of his own glory.

I've often said that Rand was misled by Moiraine into misunderstanding the purpose of the Dragon Reborn - he wasn't ever supposed to be a leader, general or king, and it wasn't until he began going into seclusion and lost his hand that he understood his role. Those two myths somewhat support an interpretation of Rand's role and future actions in that he took that role for the greater good of humanity, but will now step aside because he wasn't meant to be a worldly leader. The dragon brings chaos, destruction, salvation and rebirth. You cannot build a new order on such principles.