In The Odyssey, what does Calypso offer Odysseus if he stays with her?
The answer to this question can be found in Book V of this epic classic. After being given Jove's message from Mercury, Calypso unfortunately has to relinquish Odysseus and send him onward to his home of Ithaca and to continue his adventures. However, it is clear that she does this begrudgingly, as her response to Mercury testifies. She does go to Odysseus (called Ulysses in this translation) and tell hm that he is free to go. However, when she sees how eager he is to actually leave, she says the following words:
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 at 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.
Calypso therefore promises Odysseus the gift of immortality if he would only give up his wife and be content with Calypso and her fairer form. She also promises that his life would be a lot less eventful if he chooses this course and does not try to reach Ithaca.
Prior to now, Calypso has really kept Odysseus as her prisoner. It's a beautiful prison, certainly, but it's a prison nonetheless. However, when Calypso describes what Odysseus could have if he remains with her, she offers him the chance to be the "guardian of [her] home." This seems to imply that, if he chooses to remain there on her island, he will no longer be a prisoner as he has been, but that she will actually treat him more as an equal.
Such a promise is quite an opportunity for an immortal goddess to offer to a mere mortal man. In ancient Greece, men were certainly considered to be superior to women, but all mortals were considered to be far inferior to the gods; in fact, mortals who aspired to be godlike usually died terrible deaths in Greek myths. However, Calypso offers Odysseus the chance to act as her equal: he would guard her home and become immortal. This is no small offering.