So why do I think this shitty, comic-relief chicken is important? Because he represents the value of morality. Bear with me here.
The message of the story isn't one that has much to do with morality, it just has to do with belief in oneself. But, the writers did want to make sure that they did include something about being a good person, and that's where the chicken comes in. His name is Heihei, by the way. I looked it up.
Heihei isn't cute. He isn't smart. In fact, he's so dumb that he doesn't even really know how to eat without assistance - he likes to try to eat rocks, and needs to be pointed in the proper direction of real food. On a dangerous trip like the one he's on with Moana, he's more than just useless - he's a liability. But Moana values all life - that's part of what initially drew the sea to her - and she takes care of Heihei despite his complete lack of any redeeming qualities, and despite the fact that he does occasionally annoy her.
For almost the entirety of the movie, he's completely dead weight. He gets in the way, and honestly, he doesn't even do a great job at being comic relief to the audience. But then, at one crucial moment towards the end of the movie, and only for that one moment, he's useful. His tendency to try to eat rocks drives him to save the Heart of Te Fiti as it's about to be lost underwater while our protagonists are racing away from Te Ka, the lava monster. Heihei's moment of glory is unintentional and over in an instant, and he goes back to being absolutely useless for the rest of the movie immediately afterwards.
But that one moment is important because it shows the importance of not just looking at the most obvious, efficient way of accomplishing your goal, but instead making sure to follow one's moral values in the process. There were many times throughout the movie when Moana could have decided to ditch Heihei and would have ostensibly been better off without him, but she didn't, and in the end, in that one moment, it became her saving grace.