Why did Galileo Galilei name Jupiter’s moons after the Medici family?

Galileo Galilei named the four moons he discovered orbiting Jupiter the "Medicean stars" (Sidera Medicea) in honor of his patron, Cosimo II de’ Medici, the Grand Duke of Tuscany.12

Galileo had previously served as a mathematics tutor to Cosimo II for several summers. By dedicating his book, The Sidereal Messenger—which detailed his groundbreaking telescopic discoveries—to the Grand Duke and naming the moons after the Medici family, Galileo sought to honor his benefactor. This gesture proved successful, as it helped secure his appointment as the mathematician and philosopher to the Grand Duke of Tuscany, allowing him to return to his native land in 1610.2