There are many arguments to be made concerning Greek mythology, or any other culture's mythology, for that matter. Here are some suggestions:
1. It could be argued that Greek mythology is the basis of our literature of superheroes. This would allow you to try to trace a direct line between the gods and heroes of old and the superheroes in today's culture.
2. It could be argued that Greek mythology reflects the culture from which it emerged. What attributes of this body of literature show what Greek culture was really like?
3. It could be argued that Greek mythology has much that is valuable to teach us in today's world, about various attributes of character, for example, heroism or hubris, about love, hate, and envy. Humans have not really changed in many ways, and the stories these myths contain are arguably lessons for us still.
4. It could be argued that Greek mythology was an attempt to explain phenomena in the world that the Greeks did not understand. This is generally true of most mythologies.
5. It could be argued that science has destroyed mythology, and there is really negative literary evidence for this. When was the last time a culture added to its body of mythological literature?
6. It could be argued that Greek mythology is very different from Roman mythology. Of course, this would entail research into both.
7. It could be argued that mythology was the Greeks' text for its religion, much as the Bible is for Christians and Jews today. There is historical evidence of Greek religious practices to support this, along with plenty of archaeologic evidence.
I hope this helps you, or at the very least, inspires you to come up with an entirely new idea!