Asked for by name
When Musa was told to go to Pharaoh, part of what he asked for was a person. And appoint for me a minister from my family — Harun, my brother. Increase through him my strength and let him share my task (20:29-32).
In Surah al-Qasas he gives his reason: And my brother Harun is more fluent than me in tongue, so send him with me as support, verifying me (28:34).
Musa named the thing he was not good at and asked for the person who was. He was answered in four words: You have been granted your request, O Musa (20:36). There is no rebuke in it, and no suggestion that he should have managed alone.
That is the first half of Harun's story, and it is unusual: he is in the Quran because somebody asked for him.
Left in charge
When Musa went to the mountain for his appointment with his Lord, he said to Harun: Take my place among my people, do right, and do not follow the way of the corrupters (7:142).
While he was gone the people made a calf out of their ornaments and began to worship it. Harun did not go along with it. The Quran records what he told them at the time: O my people, you are only being tested by it, and indeed, your Lord is the Most Merciful, so follow me and obey my order (20:90).
Their answer: We will never cease being devoted to it until Moses returns to us (20:91). He was one man, and they would not listen.
When his brother came back
Musa returned angry and grieved, and took hold of his brother. Harun's answer is one of the most human lines in the Quran: O son of my mother, do not seize me by my beard or by my head. Indeed, I feared that you would say, You caused division among the Children of Israel, and you did not observe my word (20:94).
In Surah al-A'raf he says: O son of my mother, indeed the people overpowered me and were about to kill me (7:150).
He begins by calling him son of my mother — appealing to what they share before defending himself. And his actual defence is that he was afraid of making things worse by forcing a split. Musa then prayed for forgiveness for himself and his brother (7:151).
For a child, this part is useful in a way the heroic stories are not. Harun did the right thing, could not stop it happening, and was blamed anyway.