Nelson Historical Society Journal, Volume 2, Issue 1, 1966
Named after the English patriot.
Now Franklyn. At the time of naming the street, Sir John Franklin was Governor of Tasmania, and visited Wellington with Lady Franklin in March, 1841. A ship calling at Nelson in 1843 was named the Sir John Franklin. The street name occurs in Wellington and Murchison but in all cases it is spelt with an -i and not a -y.
An alternative suggestion is the name of a governor of Nelson College but that seems a bit late.