In the booksEdit
An Unsullied by the name Stalwart Shield plays the same role in the books.
Grey Worm chose to keep his demeaning Unsullied name, to defiantly reappropriate it from the slave-masters, i.e. they attempted to give him a slave-name indicating that he is worthless, but he chose to turn that around and make "Grey Worm" the name of a free man who killed slave-masters. Some of the other Unsullied, however - those who were born into slavery and don't remember their original names - do take up the suggestion to make new dignified names for themselves, such as "Stalwart Shield".
The TV series may have thought this would take too long to explain, so they renamed the character "White Rat" to keep with the previously established thematic naming of the Unsullied ("Grey Worm", "Red Flea", etc.)