This phrase was used in medical texts of the time such as Alexander Haig's Uric Acid (1904), which happened to be part of Joyce's Trieste library:
Other characters of the collaemic face are the watery, oedematous and shiny condition of the front of the eyeball, commonly called 'Bright's bright eye' because it is so often met with in Bright's disease. (p. 58-9)
Also in the Clinical Journal (1905), vol. 25, p. 81:
He had, indeed, what has been called since Bright's time, 'Bright's bright eye'.
The term 'Bright's eye' can be traced back to at least 1878 in Jean Martin Charcot's Lectures on Bright's disease of the kidneys: delivered at the School of Medicine of Paris:
In the contracted kidney there existed most frequently a little puffiness, and oedema of the conjunctiva, known by English authors as 'Bright's eye'. (Grainger-Stewart) (p. 57)
The medical term for this condition used to be 'Retinitis morbus Brightii'.
Search by keyword (within this site): Medicine France Trieste Books