Who is this phantom that appears to Penelope at the end of book four of "The Odyssey"? Is it Athena herself as the shape of Penelope's sister or is it really her sister?
It actually isn't either, technically. Athena creates a phantom in the likeness of Penelope's sister and sends it to Athena. So it isn't actually Penelope's sister, but because Athena "sends it", it also isn't Athena. Instead, it is just a created apparition, a fake ghost if you will:
Then Athena bethought her of another matter, and made a vision in the likeness of Penelope's sister Iphthime daughter of Icarius who had married Eumelus and lived in Pherae. She told the vision to go to the house of Odysseus, and to make Penelope leave off crying, so it came into her room by the hole through which the thong went for pulling the door to, and hovered over her head saying,
“You are asleep, Penelope: the gods who live at ease will not suffer you to weep and be so sad. Your son has done them no wrong, so he will yet come back to you.”
Athena will appear herself to Odysseus, but Penelope is important enough to see the real goddess, even in disguise. Instead, she gets a "note" from the goddess in a ghost disguised to be her sister.