In Homer's Odyssey, the title character and his family are supported greatly by goddess Athena throughout the epic. As a goddess of wisdom, war, and weaving, Athena can be very useful to Odysseus and his family.
As far as the reunion of father and son is concerned, we see a key moments in this event in Odyssey 13 and 15. In Odyssey 13, when Odysseus returns to Ithaca, he encountered Athena, who disguises him as a beggar and tells him to go first to the hut of the swineherd Eumaeus.
Similarly, in Odyssey 15, Athena visits Telemachus, who has been out searching for his father. The goddess meets him in Lacedaemon (Sparta) and tells him it is time to return home to Ithaca. When Telemachus arrives in Ithaca, she also tells him to go to the hut of the swineherd Eumaeaus.
Thus, having instructed father and son to be in the same place, the stage is set for the tearful and joyous reunion in Odyssey 16:
"With this he sat down, and Telemachus flung his arms around him and wept, and the longing to express their emotion rose in both their hearts. They keened aloud, and their cries rose, louder and more frequent than those of birds of prey with curved talons, vultures or sea-eagles, whose nests have been robbed of their unfledged chicks by country folk. Tears streamed piteously from their eyes. And sunset would have found them still weeping..." (A.S. Kline translation)