ASOIAF: Bran is more like Frodo in ASOIAF than any other character, yet unlike Frodo, the goal of his mission is unclear. What is the goal of Bran’s mission?

Bran is not like Frodo.

Bran is a 7–8–9 year old child who who has been crippled but dreams of being a great knight but will settle for being a wizard. His prime motivation is gaining able-bodiness, self-empowerment and agency.

Frodo is a 52 year old adult bachelor who hangs out with twenty-somethings, because they don’t mind his weirdness, likes hiking and languages and dabbles in poetry.
He is highly intelligent; fantastically rich but unambitious; his motivations is a sense of heavy duty — which he got because he is essentially free from an urge to power and thus uncorruptable.

Bran has a lot of self-pity because of his injury; Frodo, on the other hand, pities everybody else he sees: Boromir, Gollum, even the Orcs.

Bran, like most characters in A Song of Ice and Fire, is seeking power at some level; Frodo is trying desperately to get rid of power.

These are majorly different themes in the two stories, by the way.

The only correlation between the two of them is that they are journeying into danger, and fleeing from danger. And they have a broad hero’s journey with mentors and so on(but again, Frodo being a grown man — er, hobbit that is— changes the mentor -pupil dynamic. He is not really a padawan learning about life the way Luke Skywalker or Bran is.)

AS to what Bran’s mission is: I think — and there is some ambiguity, I’ll admit, in the books — it is mainly to take his place as a Greenseer, i.e., an Old God, a protector of those who pray to the Heart Tree…and to see the Winter through to it’s conclusion to Spring, which means defeating, or playing a role in the defeat of the Others.

IN the sense that Frodo and Bran are serving all humanity, I suppose they are similar. But they are very different characters, with different roles and different goals.