Texan and Dallasite are the most common terms. Dallas native is another one. Even being a Coloradan for eleven years along with living in Germany and Croatia for a good while hasn't stopped people calling me a Texan. As stated previously, though, Dallasite is pretty common, too.
Why is Texas called the "Lone Star State"? Texas's nickname pays tribute to the Lone Star flag, which was adopted after Texas became independent from Mexico in 1836.