From the Super Mario Wiki
The Horse's Bit, also referred to as simply the Bit, is an item in the SNES and PC versions of Mario's Time Machine. A bit is a device that is placed in a horse's mouth, which lets the rider control it. This bit belongs to the Duke of Alençon's horse, who had it on his possession during the battle at Orleans. When Mario travels back in time with Joan of Arc's Shield, he can give the Duke a Quill Pen; the Duke then asks Mario for a favor and gives him the Horse's Bit, telling him to bring it to the stablemaster. The stablemaster had previously mentioned that he needed to fix the bit, and when he receives it, he tells Mario that he saw the hoofprints of Joan of Arc's horse heading up a hill on the right, which lets Mario know where to find Joan.