Assording to legend StBernard's Well was originally discovered by Saint Bernard of Clairvaux, the founder of the Cistercian Order, in the 12th Century, who was ill and living in a cave near the Water of Leith. He noticed the spring because so many birds visited it and he drank its healing waters until his strength returned.
The well was re-discovered by three school boys from Heriot's School while they were fishing in the Water of Leith in 1760. A simple well house covered it until 1789, when the current building was commissioned by Lord Gardenstone and designed by Edinburgh landscape painter Alexander Nasymth. The marble statue represents Hygieia, Goddess of Health.
For decades wealthy holiday makers used to visit Edinburgh to take the well's waters. These waters were believed to cure arthritis, back ache, and even blindness. The taste was apparently pretty disgusting though. At one point the pump room apparently resembled a continental cafe and the water was even bottled at one point. The well remained open until 1940. Nowadays you can access the interior only on special occasions such as Edinburgh's Doors Open Day.
I posted a photo of bridges further up the Water of Leith over on Crafty Green Poet. You can see them here.