Unfortunately, your information is faulty.
According to the book The Franklin Comes Home (Hawthorne Books, 1974) by A.A. Hoehling, written about the catastrophic damage the USS Franklin received on 19 March 1945, the ship was named for the Battle of Franklin, TN from the Civil War and NOT for a ship of the line or for Benjamin Franklin. This is printed on page 3.
Yes, I'm aware of that claim, it has obviously been around for a long time. But the Dictionary of American Naval Fighting Ships is the US Navy's official word, and it clearly states "The first four ships of the name honor Benjamin Franklin; CV-13 perpetuates the names of these ships," as Bloodwhiner has already cited.
The reason I referred to the ship-of-the-line Franklin (Franklin no. 3) as the namesake is that none of the others were particularly noteworthy. As one of the first line-of-battle ships commissioned in the USN, "Franklin" became one of the service's "prestige" names. The others were Independence, Columbus, Washington and Ohio. "Independence" was also used for a carrier; the others having corresponding place names would obviously go to battleships or cruisers before a carrier.
--Justin