What do you call someone from the Sun?

Next to "Solarian," you might want to consider Helian.

HELIAN

from Greek helios (Sun) + suffix -ian

A learned synonym for the term solarian - it is derived from the higher-status Greek, not from the lower-status Latin. The term is indicative of its user's erudition. Digital Commons @ Butler University

heliacal: relating to or near the sun M-W

So what should you call someone from the earth? Tellus was the goddess of the earth, which gives us Tellurian (the preferred choice of E.E. "Doc" Smith). The Greek counterpart is Ge or Gaea, from which we get words like geology and perigee. A person living on Gaea would be a Gaean.

"Terra" (and consequently "Terran" and "terrestrial") is seen a lot in SciFi, but it's no more a proper name than "earth" is. "Terra" is simply the Latin word for "dirt" or "land." It's not the name of a god or goddess, so it doesn't follow the rule for the names of the other planets.

"Earthling" is awfully retro and "Earthian" is just beneath contempt.

Unlike the earth, the sun does have a name: Sol (with the Greek equivalent being Helios). So an inhabitant of the sun would be a Solarian or Helian. io9.com

Another possibility, by analogy with H. G. Wells' "Selenite," is Solarite.

I think Solarling or Sunling would be interesting, seeing as we are called "Earthlings". It'd be a nice name for a race a bit on the "cuddly" or more "familiar" side, though, as the name gives the connotation of that- which I doubt is what you are going for.

starchild or children of the sun could be for what you are looking for. It is a fact that our sun is a star by its astronomical definition.

An alternative could be found in the lore of the SF show Andromeda. As a avatar of the suns, Trance Gemini was a sun-born lightbringer, a immortal humanoid form of a star that co-exist in the common spacetime.

So, starchild or sun-born / solar-born are the closest definitions from my point of view.