They were originally tea bowls which got very popular by Buddhist monks who drank tea as a way of keeping themselves awake during their nightly ceremonies during the Song dynasty. According to tradition this Tianmu mountain was the site of a Buddhist temple visited by Japanese Buddhist monks who took these bowls back to Japan. A few of these old monk bowls are believed still to exist in Japan where they are individually named and kept as national treasures. Regarding the value of your bowl I figure the personal attachment to a piece with a family history by far exceeds its monetary value. This information is given as an example of private conversation only and is not intended to be used as a promotion of any individual piece.
All opinions is the authors and is given as such with all hazards of judgeing anything from a photo.