Red hood
In short: The gift from her grandmother that gives Little Red Riding Hood her name.
Object
How does it work in the tale?
In Perrault and Grimm, the hood makes the girl recognisable. The wolf’s disguise works in reverse: the grandmother’s appearance hides who is really in the bed. Dress serves both recognition and deception.
When does a cap become a name?
In the Grimms’ telling, the grandmother gives the girl a red velvet cap. She wants to wear it all the time, and the narrator explains how that habit gives her a name. When the wolf meets her in the woods, he uses that same name to greet her. Before the girl sets foot in the woods, readers have already seen how a gift becomes the name others call her by.