The term dream diary was coined by Ann Faraday who was a psychology student and researcher. The study shows that she spent time in India and other Asian countries as well before finally setting in Australia. She came in touch with Zen activities and Yoga while researching on dreams.
Dream diaries can be helpful for tracing down your sequence of dreams and to interpret them to get visions and premonitions of the future according to some practitioners.
However, the practices are running up since the ancient time.
There are the evidences people have found in Roman and Indian Texts that show us similar ancient practices in these civilizations.
Resources: Introduction to dream diaries