In the Odyssey, what is the most important incident in Telemachus' life that made him a man?
There are a number of different events where you see Telemachus moving toward manhood. He stands next to his father when going to war with the suitors, he stands up and insists that the "beggar" be treated well when the suitors are acting poorly, he sits down and has a man to man talk with Menalaus. All of these show that he is leaving the realm of childhood behind and starting to interact with others on an adult level.
However, I would say that the most important incident was his decision to leave Ithica and go searching for word of his father. Before this incident, Telemachus was a child within his mother's household. By making the decision to leave, he was clearly showing that he now saw himself as a man who had the right to make his own choices. Athena then sanctified this choice by giving him advise and support. The moment he decided to take a ship and leave, the suitors knew they had to get rid of him because that choice made him a man. The later incidents simply showed what a good man Telemachus was turning into.