Who are the characters that would be considered friends?

Certainly, Odysseus is friends with Menelaus.  In fact, it was to recover and protect Menelaus's honor that Odysseus went off to fight in the Trojan War in the first place.  When Paris, the prince of Troy, stole Menelaus's wife, Helen, hers became the "face that launched a thousand ships," according to the playwright, Christopher Marlowe.  Then, when Telemachus travels to Sparta to see if Menelaus has any news about Odysseus, Menelaus tries to be as helpful as possible out of loyalty to his old friend.
It seems as though Odysseus and Agamemnon, Menelaus's brother, must have been friends as well.  When the two meet in the Underworld, Agamemnon attempts to give Odysseus some friendly advice about how best to handle his wife, Penelope.  Agamemnon is still pretty bitter about being murdered by his wife and her lover, and he seems to want to protect Odysseus from being betrayed himself.
In addition, Athena seems like a real friend to Odysseus.  She intercedes with Zeus on Odysseus's behalf, she helps him to find the assistance he needs in Phaeacia, she helps him vanquish the suitors, and on and on.  She even helps his son to begin to make his own name in the world, protecting him from harm, and comforts Penelope when she is upset.