That original trophy is now in the Baseball Hall of Fame, put there in 1988 after 55 years of traveling from city to city.
The idea for the current IL playoff format came from Frank Shaughnessy, better known as "Shag". He presented his idea to the league in 1933 and it was adopted. Shaughnessy would go on to serve as President of the International League from 1936 until 1960, but while he was a league president, he was never a Governor.
To find out how the trophy got it's name, you do what you do most times to find answers like these; you follow the money. The trophy was sponsored - paid for - by the Governors of New York, New Jersey and Maryland and the Lieutenant Governors of Quebec and Toronto. That covered the reach of the International League teams geographically and the Governor's Cup was born.
While the trophy may look rather ornate, it's actually not as valuable as you may think. The value of the original trophy is estimated to be just over $3000.
A new trophy, bearing the same name and style, was given to the league champion starting in 1989. The new trophy has traveled to some of the same cities that the original one did, and a few new ones like Richmond, Durham, Toledo and Louisville. For the time being, it resides in Columbus, Ohio, home of the Columbus Clippers.
For the Lehigh Valley IronPigs, it seems convenient enough that they should be able to just bring the trophy back to Pennsylvania with them. After all, it's spent just one season in the Commonwealth, when the Scranton Wilkes-Barre Yankees won the cup in 2008.