A Game of Thrones
Anything that breathes fire is worth talking about, we think. Along with the weird long seasons (and food), dragons are probably the most magical symbol we have. In fact, that's a lot of what dragons symbolize: they're a reminder of magic. More specifically, since they're extinct (cough cough), they're also a symbol of the lack of magic in the world. Like Valyrian steel and the children of the forest, dragons are a reminder of a more magical past.
But just one fire-breathing second: dragons might not only be part of the past. Both Tyrion and Arya have strange encounters with the dragon skulls kept in King's Landing, where the dragon skulls don't seem totally dead (14 Tyrion 2.21-2, 33 Arya 3.27). And now that you know how the book ends, you know that dragons aren't really extinct: they were just sleeping until Daenerys woke them up by hatching those three seemingly-fossilized eggs. Crazy.
Even though Viserys often threatened his sister by telling her that she didn't want to "wake the dragon," Daenerys becomes more powerful only after she accepts this legacy of the past and figures out how to awaken it in this present. Perhaps then, dragons are a symbol of power (and Daenerys' power especially), and also a reminder that we can't always be sure who has or will have that power. After all, someone who doesn't have power at one moment (Daenerys after Khal Drogo dies) might discover some new source of power in the next moment (Daenerys after she hatches her dragons).
So, like the Iron Throne, dragons remind us that power can shift over time. That makes some sense, since the Iron Throne was made in dragon-fire. Well played, Martin, well played.