"short, thick stick of wood used for fuel," mid-15c., from Old French billette, diminutive of bille "stick of wood," from Medieval Latin billia "tree, trunk," which is possibly from Gaulish (compare Irish bile "tree trunk").
The sense development in French perhaps is from small cards written or printed with instructions for how to behave properly at court (compare Italian etichetta, Spanish etiqueta), and/or from behavior instructions written on a soldier's billet for lodgings (the main sense of the Old French word).
also fiddlehead, "one with a head as hollow as a fiddle," 1854 (fiddleheaded), from fiddle (n.) + head (n.). As a name for young fern fronds, from 1877, from resemblance to a violin's scroll. Earliest use is nautical, "carved ornamental work at the bow of a ship in the form of a scroll or volute" (1799).
There are three kinds of heads,—1st The Figure-head is one on which is placed the figure of a man, woman, or the like, &c.; 2d, The Billet-head, or Scroll-head is one finished with two scrolls or volutes ...; and 3d, the Fiddle-head, which is finished with only one scroll or volute, having the spirals turning inwards to the vessel. [Peter Hedderwick, "Treatise on Marine Architecture," Edinburgh, 1830]