If his appearance has captivated most painters, what has most intrigued them about Lord Ganesha is his vehicle, the tiny mouse.
There are many stories about how such a creature came be the elephant god's mount.
One such story says many people came to Lord Shiva and Parvati's house to see their new child. Everybody gave some gifts. The Earth gave a mouse as Ganapati's vehicle.
Also See: The Ganesha collector