Derived from the Germanic element id meaning "work, labour". The Normans brought this name to England, though it eventually died out there in the Middle Ages. It was strongly revived in the 19th century, in part due to the heroine in Alfred Lord Tennyson's poem 'The Princess' (1847), which was later adapted into the play 'Princess Ida' (1884) by Gilbert and Sullivan.
Though the etymology is unrelated, this is the name of a mountain on the island of Crete where, according to Greek myth, the god Zeus was born.