The Portuguese arrived in India in 1498 and began evangelizing the local population. They converted the people of north Konkan --Bombay, Bassein, Thane etc and called them: "Portuguese Christians".
Later, the East India Co came to India in 1600 as traders, mainly in spices. Spices were important since they were used as preservatives, especially for meat, as there were no refrigerators. In 1661 the Portuguese gave Bombay as dowry to the British after King Charles II of England married princess Catherine of Portugal. A little later, in 1668 Bombay was given to the East India Co for an annual rent of £10.
With the advent of the railways and steam ships, immigrants from Goa began coming to Bombay for jobs. They lived in clubs known as kudds.
The local Bombay Portuguese Christians wanted to distinguish themselves as the sons of the soil, from the Goan Portuguese Christians, so that they don't lose out to the Goan immigrants in jobs. So during the Golden Jubilee celebrations of the Coronation of Queen Victoria in 1887 they made a request to the Empress for them to be known as East Indians. It is not known if Her Majesty acceded to their request but from that time (1887) they have called themselves East Indians.