In The Odyssey, what means does Homer use to show the strength that characterises the family of Odysseus, Penelope, and Telemachus?
Homer employs epithets, words or phrases that characterize a person which accompany their name, to draw attention to the particular strengths of Penelope, Telemachus, and Odysseus. For example, Penelope is often described as "heedful Penelope," emphasizing her sense of duty to her husband and son as well as the strength of her will to listen and obey (qualities highly prized in women). Telemachus is often "discreet Telemachus," and his epithet recognizes his strengths: discernment and modesty. He exercises good judgment and is not boastful or proud, despite his social standing. Finally, Odysseus is often described as "royal Odysseus" or "princely Odysseus," epithets that draw attention to his royal mien and princely nature.
Further, Homer makes it clear that this family is one to which his audience should look up by how willing Athena is to intercede with and help all three of them. That a goddess would involve herself so heavily in the affairs of just this one little family seems to show that their strengths, outlined above, are impressive indeed. The gods do not typically intercede with mortals who they see as weak or impious, and so Athena's willingness to help the three of them says quite a bit about the respect accorded them for their many strengths.
You might like to consider the way in which these three characters are depicted throughout this epic classic, and in particular the way in which the bonds of their family are so strong that they continue to persevere in their hope that they will be reunited one day. Consider Odysseus. What drives him home through all of his trials is the thought of being reunited with his beloved Penelope and his son Telemachus. She is his ultimate motivation. Secondly, Telemachus is depicted as a loyal son, doing what he can to defend his mother against the suitors and also helping his father in every way when he does return to Ithaca. Lastly, Penelope has become a byword for a devoted and faithful wife. Penelope comes up with various stratagems to resist the advances of the suitors so that she is not forced to pick one of them. Her love and faithfulness towards her husband is clearly evident. It is through such characterisation, therefore, that Homer shows the strength of this family unit that is able to withstand vast periods of separation and still remain undefeated.