Why does Athena play a key role as a mentor for Telemachus?
This is a great question, because as the Odyssey begins, Homer assumes a lot. Without knowing about the Iliad and other stories of Greek mythology, there can be confusion, such as why Athena helps Telemachus.
Within the Greek world, all the gods have their favorites. Athena does as well. For her, Odysseus is her favorite. Part of the reason for this is because Odysseus has characteristics that are like hers. For example, Athena was a strategist of war (not a destroyer, like her brother Ares). This resembles Odysseus. The artifice of the Trojan Horse is an example.
Another reason why Athena looked upon Odysseus with favor is because Poseidon hated Odysseus for what he did to Polyphemus. Athena and Poseidon have a rivalry. Hence, by helping Odysseus, she is also going against Poseidon. An enemy of my enemy is my friend.
In view of these points, we are now in a position to answer why Athena helped Telemachus. Athena helped Telemachus, because she wanted to help Odysseus.