The quest or pursuit for immortality remains a recurring theme in many Greek stories. Immortality was highly valued by the Greek Heroes. Mortals would always look for ways to become immortals and avert death forever. Immortality was associated with divinity and all Gods were considered to be immortals. Being mortal meant to be a human and someone who could die.
In Homer’s Iliad, it is the Greek warrior Achilles whose pursuit of immortality is best remembered. Achilles is given two choices that will decide his future. He can either go to fight in the war or stay at home with his family. If he chooses to stay back with his family, he will live a good and peaceful life, but will not be remembered. This will be the plain life of a mortal. But if he chooses to participate in the war, though he is destined to die, his death will be a glorious one. He will be forever remembered for his courage and excellence. As we see, Achilles chooses to die in the war and exist forever. This is becoming immortal by being remembered forever.
My mother Thetis tells me that there are two ways in which I may meet my end. If I stay here and fight, I shall not return alive but my name will live for ever: whereas if I go home my name will die, but it will be long ere death shall take me
In Homer’s Odyssey, Odysseus is also presented with a choice to decide his future. But his decision is very different compared to Achilles'. Odysseus declines Calypso’s offer to marry her and become immortal. He is more interested in uniting with his family and loved ones than becoming immortal. It is shown that he is not afraid to live and die as a mortal. Calypso mentions how important and prized immortality is.
Good luck go with you, but if you could only know how much suffering is in store for you before you get back to your own country, you would stay where you are, keep house along with me, and let me make you immortal, no matter how anxious you may be to see this wife of yours, of whom you are thinking all the time day after day; yet I flatter myself that I am no whit less tall or well-looking than she is, for it is not to be expected that a mortal woman should compare in beauty with an immortal.
We see that Odysseus does not consider immortality to be so warranted as does Achilles. Achilles chooses for an everlasting "life of fame" after death. But Odysseus chooses to live a happy, mortal life with his wife rather than become immortal. So the attitude towards immortality shows a change from Iliad to The Odyssey.