The Abura-akago is a "flaming" baby that is said to steal the oil from lamps by licking it up with its long tongue.
According to Toriyama Sekien's explaination[link]
long ago in the village of Shiga there was an oil thief who, every night, stole the offerings of oil from an OJizou-sama (the patron saint of travellers and guide for the souls of deceased children) statue which stood at the crossroads leading into the village. When this thief died, as a form of punishment, his soul was transformed into a wandering flame, however, others believe that the Abura-akago is the reincarnation of the thief who continues to want to steal oil.
Because the Abura-akago is a ghost child and is associated with Ojizou, it reminded me of Osore-zan in the north of Japan where, traditionally, people thought the entrance to Hell (the afterlife) was located. Even today, many people make a pilgrimage to Osore-zan in the summertime to say goodbye to or communicate with the deceased (via blind mediums known as Itako), but, more specifically, couples who have lost a child will often place pinwheels at Osore-zan for their kids to play with in the next world: [link]
So, I decided to give my Abura-akago a pinwheel to play with, too.