"Twila" is an alternate spelling of the name Twyla, which is in this site's database. (click on blue hyperlink). The form of "Twyla" is perhaps the best known, due to having been made popular by the choreographer Twyla Tharp.
Of uncertain origin, it may have been H.L. Mencken who suggested that this name -- first used among Black American families in the Southern United States -- was an invention using the word "twilight" for a child born at sundown.
Others have suggested that the name springs from the French word "etoile", meaning "star"; while still others suggest that the name may have come from the word "twill" (a kind of fabric).