NotGraphs Sponsors a Baseball Reference Page

It goes without saying that one cannot put a price on laughter.
The price of NotGraphs’ first Baseball Reference sponsorship, however, is two dollars annually.
Carson Cistulli has published a book of aphorisms called Spirited Ejaculations of a New Enthusiast.
Sloppy Thurston was my favorite baseball player name ever. (http://www.baseball-reference.com/players/t/thurssl01.shtml)
Until now.
So would his ballplaying namesake son be Sloppy Thurston Jr., Sloppy Thurston II, or Sloppy Seconds Thurston?