Why is there a “riding” in “Little Red Riding Hood”?
(a) A hood formerly worn by women when riding.
(b) A kind of cloak with a hood.
The original German title of the fairy tale, "Rotkaeppchen", has no connotations with riding gear. This only got in when it was translated into English. The literal translation of the fairy tale title would be "Little red hat" (or cap).
A reasonable explanation would be that the Riding was added to form an alliteration, making it sound better and more memorable.