In the comic books, Green Arrow operated out of Star City. But in the new 'Arrow' series from The CW, Oliver Queen lives in Starling City.
It could be that it was originally called Star City, but the city decided to rename itself after a major benefactor, Henry Starling......
From Memory Alpha, a 'Star Trek' wiki:
Henry Starling was a Human industrialist who was largely responsible for the computer age of the late 20th century. Unknown to anyone at the time, Starling's innovations were based on his primitive understanding of the technology found aboard the 29th century timeship Aeon, which he appropriated after discovering the crashed vessel in 1967.
(Henry Starling appeared in two episodes of 'Voyager': "Future's End" and "Future's End Part II".)
Because of the innovations he introduced (rather than invented), Starling became a man of wealth and power throughout the world. And he won many philanthropic awards over the years. As part of his philanthropy, he may have contributed millions of dollars towards the revitalization of Star City (even though he lived in Los Angeles) and in thanks the city changed its name to honor him.
I don't think the truth about Starling ever surfaced in the present day of Toobworld, and his death was probably covered up by temporal agents from the Future. So that's why Starling City would still bear his name more than fifteen years after he was killed.
This is all speculation of course. But it works for the Toobworld Dynamic.....