In Greek mythology, for example, Apollo could be considered in some ways a god related to stories. He was the god of poetry, among other things, and poetry is a form of storytelling. Another one could be Hermes, who was a messenger and often carried tales from the gods to mortals.
In Norse mythology, Odin is often associated with stories. He was the All - Father, and he hung himself on the world tree to gain knowledge, including the knowledge of runes which were used to tell stories and hold wisdom. His ravens, Huginn and Muninn, also flew around the world and brought back stories to him.
In Hindu mythology, Saraswati is the goddess of knowledge, music, and the arts. Since stories are an important part of art and knowledge, she can be seen as having a connection to the idea of a 'god of stories'. Her influence on literature and the spoken word makes her relevant in the context of stories.