What does it take to convince Telemachus of his fathers identity?
I think that you are talking about what happens in Book XVI.
Odysseus is disguised as a beggar at the beginning of this chapter (Athena has disguised him). He has been hanging out in the hut of Eumaeus, the swineherd. Telemachus comes to see Eumaeus. After a while, Athena decides that it is time for Odysseus to reveal himself to his son and she takes off the magic disguise.
But even when she does this, Telemachus thinks Odysseus is a god. What it takes is for Odysseus to tell Telemachus that it is not strange that he should be there -- he tells his son that Athena was the one who disguised him and changed him back. That convinces Telemachus, though I'm not sure why it would...
Here's the passage:
What you wonder at is the work
of the redoubtable goddess Minerva, who does with me whatever
she will, for she can do what she pleases. At one moment she
makes me like a beggar, and the next I am a young man with good
clothes on my back; it is an easy matter for the gods who live
in heaven to make any man look either rich or poor."
As he spoke he sat down, and Telemachus threw his arms about his
father and wept.