Telemachus passes an important test in Book I with Athena, and the suitors fail the test. What does his behavior in contrast to the suitors tell us about what Homer's culture valued most?
The original question had to be edited. I think that one of the most telling elements out of the contest in Book I is that it shows the difference between the contingent and permanent. The Greek society of Homer embraced individuals who were able to see through the former and focus on the latter. It is for this reason that heroism on the battlefield, arete, was worshipped because it embodied the transcendent. It is also for this reason that cowardice and reverence for the contingent are frowned upon in Greek society.
Telemachus passes an important test in following the path set out to him by Athena. In this, he shows his embrace of the permanent and the transcendent. As the suitors show their embrace of the contingent in seeking to win Penelope's hand and Odysseus' kingdom, the reader is reminded as to how embrace of the contingent is frowned upon in Greek society. When Telemachus dismisses the suitors and begins to embrace the word of Athena, it reflects the value that Greek society places on the power of the transcendent. Such a moment also details how the society of Homer frowns upon those who only live for the temporary.