I've never really thought about it before, but I'm not sure if "The Sun" actually has an officially sanctioned name! I'll accept "Sol", and "The Sol System". I'll also accept "Helios/The Helios System", or the "Terran System" to refer to our entire Solar System. And I'm sure many different cultures have their own term. But in terms of science/academia, I'm not sure that it has an official name. Looks like the IAU (International Astronomical Union) hasn't given it a sanctioned designation. [0] [1] [2]

[0] https://earthsky.org/space/what-is-the-suns-name/

[1] https://www.iau.org/public/themes/naming/

[2] List of IAU Solar System Objects, Exoplanets, Starts, and Stellar Streams https://docs.google.com/spreadsheets/d/1CEXGyancLRtHyPW7u0L_...