One of the themes of the Odyssey is the difficulty of homecoming. This presupposes one thing - the fact of being away. So, anyone who has been away and who longs to come home fits at least one theme of Homer's work. Immediately what comes to mind is the amount of mobility there is in the modern world. In the past, people lived in one town and died in one town. Now people are so mobile that they live far away from family and many things may keep them there. So, if people in this situation desire to come home, there is something in the Odyssey for them. The most obvious example will be a reference to the many wars that were fought in the modern world.
You need to remember that The Odyssey straddles the line between legend and myth. Though scholars generally agree that there was once a city of Troy that was conquered by Argive forces (which places The Odyssey in the realm of legend), clearly many of the episodes are fantastic, making it mythic. To attempt to "compare" modern events to ancient legend (or myth, depending on your perspective) is to devalue the poem's meaning and trivialize its impact.
What is important to remember is that The Odyssey is timeless. War and conquest, the desire to return to one's home, loyalty to family, and adherence to a strict code of conduct are more concepts lasting than mere historical episodes.
However you answer your question, you need to consider seriously whether it is the event or the telling which is of most importance.
A strange and fascinating question. Can I include brief references to the Trojan War as well?
The twentieth and early twenty-first centuries are marked by a number of movements: development of state power, development of advanced technologies, violence and mass slaughter, mass population growth, new explorations of the world, rising spirits and loss of spirits.
You could link most of those to the Odyssey. Many warriors went to war. The wars took longer than we thought, and some warriors took decades to get home. (Some Vietnam era soldiers are still MIA.) Immigrants must take years, and strange voyages, to find new homes—think of the boat people. Those are like Odysseus.
Many face challenges they cannot survive through force. I think of the Jews trying to slip past Nazis to survive the Holocaust; they could be fighting Polyphemus.
The lotus eaters compare to mass drug use in society—and to mass television watching.
Our leaders must steer courses through many threats on all sides, like Scylla and Charybdis.
Astronauts went to strange and magical places, like those Odysseus found on his voyage.