Likely named after Jonathan Swift (1667-1745), Dean of St. Patrick's Cathedral, Dublin, and celebrated author of Gulliver's Travels (1726). Oddly, Swift Street was not discussed by John J Marshall in his articles on Belfast street names.
"Resolved that the following names of streets submitted by the Surveyor be approved of: … Mr Coates’ street parallel to Woodstock Road — Swift Street". (15th April 1874).