The Odyssey begins with the son because the poem wants to show Odysseus's house before showing Odysseus himself. The first four books, called the Telemachy, are concerned with Ithaca and Telemachus's journey, and only in Book V does the hero appear, held on Calypso's island. Until then Odysseus exists as an absence: in Athena's complaint, in the bones that the son imagines rotting in the rain, in the song of the bard that makes Penelope weep.
The first reason is plot. Telemachus is a young man who does not yet act. In the first book he sits among the suitors, sad, imagining his father arriving to drive them out, and Athena, disguised as Mentes, tells him that he is "no longer of that age" for childish things. The goddess tells him to call the assembly, send the suitors away, and seek news of his father in Pylos and in Sparta; then she departs flying like a bird, and the young man realizes he has spoken with a god. Later, in Book XX, he will tell the suitors: "I was still a child before."
The second reason is the measure of value. Nestor and Menelaus, when they receive Telemachus, tell what happened to the other heroes on the return from Troy, especially to Agamemnon, murdered at home by Aegisthus and avenged by his son Orestes. The poem uses this case as a mirror: a son who defends his father's house earns fame among those to come, and Athena herself cites Orestes to Telemachus. It also makes clear what is at stake in Ithaca before Odysseus arrives: a house being devoured, a mother who neither refuses nor accepts marriage, and a people who are moved in the assembly and decide nothing.
Finally, the device shows that the return is not only of one man but of a family. When father and son meet, in Book XVI, Telemachus has already traveled, faced the suitors in public, and escaped the ambush, and so he can be an ally, not merely an heir. The Telemachy is thus the first half of a two-part montage: the son builds himself while the father is rebuilt.