Fimbria, direct from the Latin fîmbria, the ornamental fringe on the borders of clothing, cloth napkins, etc.
The fringe on the border of the uterine tubules were first called fimbria in the 17th century by the Dutch anatomist Reinier De Graaf (the man who coined "ovary" and discovered that ovaries produce eggs, and in whose honor the Graafian follicle was named).
Julius Caesar favored fimbria on the sleeves of his tunics (interesting as this was considered an exclusively feminine affectation by both the Greeks and Romans).
An engraving of a Roman dinner napkin with fimbria; from a painting found in Pompei.
From A Dictionary of Roman Antiquities, by Antony Rich, Appelton, 1874.