Question: why does one of the central figures in the Odyssey, the protagonist of its first four books, the son of a great hero, and a (not quite) man favoured by Athena act like a whining child?
Answer: because he is, and under the circumstances, you would be too.
The Odyssey is about fathers and sons, and so we open on four books describing a son. A son, who is in fact defined by his lack of a father. His name refers to the far off war, the circumstances that took his father away from him. He grows to the age of a man without a father, living in a house with no male role model. Instead, he has his mother’s suitors to contend with, and is thus, in every sense of the word, emasculated. And it shows.
Book Two of the Odyssey sees Telemachus deliver a speech to a council of elders, ostensibly to ask for their aid with his circumstance, but with the narrative purpose of establishing just how childish he is at this point. Compare the average hero elsewhere in Homer and Greek mythology in general. We see them fight and kill, and speak with great eloquence. But Telemachus delivers a disjointed, unpolished speech, in which he says that he “cannot fight against them” and would be “useless” (as translated by Wilson), followed soon by his breaking down into tears. He points out that he has “had no training” – and therein lies the heart of the issue. He is a child without a father, and therefore without the qualifications and training he needs to take up his role not just within the story to banish the suitors from his home, but within the established social constructs of the Odyssey and Greek society as a whole as a strong ruler.
The Telemachy, then, is the story of Telemachus’ journey not so much to find Odysseus – we see later in the narrative that it achieved nothing in this regard – but to find himself, or rather, the man he is supposed to be. Athena, the female goddess of the traditionally male domain of war as well as of wisdom disguises herself very intentionally as the male Mentor (the name being the source of the modern usage of the word mentor) and becomes the missing father figure to Telemachus. The journey offers various “teaching moments” – Telemachus learns to make the proper offerings and libations to the gods with Nestor in sandy Pylos, and he sees xenia, or hospitality, in both the halls of Nestor and Menalaus, among other things. The Telemachy is the crash course, the Rocky training montage that Telemachus takes before returning with his father, finally becoming a man and joining the grand tradition of Homeric heroism by violently slaughtering those who wronged him.
The larger context of the Odyssey is about a man’s winding, difficult journey to return from a war he did not care for, back to his wife and son. How hollow, then, would the resolution have been without the Telemachy? What if Odysseus had returned to the same whining child as his heir to the throne of Ithaka? Telemachus did not find his father – in the end, his father found him, but he had in the meantime grown as a man, and perhaps that is the point.