Do you mean a story written FOR that child about something in that childs life? I think that is a beautiful idea!
I'm not sure how much I would pay for it... it would depend on how long the story was, how the book was 'made' (e.g nicely bound, spiral bound or just pages) and what illustrations were included. I think a childrens book/story of any type needs illustrations (the icing on the cake would be nice drawings of the child in the story!), whether that is just pencil drawing, coloured drawings or photographs. I'd be pretty happy having something like this emailed because then I could choose how best to present it (probably having it bound or spiral bound).
I think done right you could create a really lovely heirloom for a child to keep forever.