It's not surprising that the Odyssey is full of suffering: its characters live in a world without antibiotics, painkillers, and iTunes. (They actually have to get in a car and drive to a store to buy the latest One Direction album.) That's just the curse of mortality. And there's only one way to deal with it: endure. But they're not particularly keep-calm-and-carry-on about it—they may put up with the suffering, but they also weep, grieve, and lament. A lot. And one odd thing: you'd think that people who thought so much about daily suffering would come up with a better afterlife for themselves. Instead, our glimpse of the Underworld makes it look like a lot more of the same.
In the Odyssey, suffering always comes from bad actions. If you could manage to live without angering the gods, you'd be able to avoid suffering entirely.
In the Odyssey, suffering doesn't serve a purpose: it's a senseless burden that all mortals must bear.