Within my Atlantis series, I always refer to the sun as Helios. The reason is I wanted to make my culture stand out from typical Earth. In other words, if my characters all swore in the same manner and called everything by the same name, then what would be the point in switching to another planet? They should have cultural differences. The easiest cultural difference would be to recognize their Greek background.
Helios was actually an easy switch. In classic mythology, Helios was one of the Titans (these were part of the original gods before the Olympians came around). Their jobs were a little more simplistic than the Olympians, which seem to span every aspect of life. I always liked the idea of Helios. He rode a chariot across the sky driven by six steeds with fiery wings. His job was to start from his home in the eastern ends of the earth, ride across the sky into the land of evenings and into a fiery bowl that would transport him back to his home.
What I find interesting about Helios is that his job is very methodical. Besides the steeds with fire for wings, his job is ordinary. Yet, even as simple as it sounds, his job is one of the most important. I think this applies to "real" life. Some jobs are mundane and boring. In fact, I know I wanted to quit many because they were not interesting enough. But the backbone of companies or even society are built upon these jobs. We cannot function without them. Therefore, Helios is one of the most important gods, even though he does not have a place on Mount Olympus.