Some elementary schools hold a time-capsule ceremony: children write "a letter to my future self" and seal it away. The fact that children write those letters with such care suggests they have an intuition that their future selves will want to know what their present selves were like. That intuition is not far off from what developmental research describes.
The study of autobiographical memory has been gradually clarifying how memory and the self develop. Those findings also offer clues as to why the period around age 8 can be a reasonable time to hand a child a diary of her own.