Defictionalization: Bubba Gump Shrimp Company. In a sense. The movie's Bubba Gump was a shrimp extraction concern, the defictionalized Bubba Gump is a chain of casual dining restaurants.
Gary Sinise and the Lieutenant Dan Band.
Development Hell: The adaptation of the book sequel, Gump Co. (helped by the fact that due to Hollywood Accounting, author Winston Groom didn't get a dime out of the $676 million the film took worldwide).
Font Anachronism: The letter from Apple Computer is dated 21 September 1975. The font used on the logo, Apple Garamond, wasn't used by Apple until the launch of the Macintosh in 1984. The font was only designed in 1977.
For that matter, the rainbow Apple logo was first used in May 1976.
Kurt Russell does a quick cameo voicing Elvis Presley. It wasn't the first time (It Happened at the World's Fair), the second (the TV biopic Elvis directed by John Carpenter) nor the last time he played the King.
I Am Not Spock: Gary Sinise became so strongly identified as Lieutenant Dan that he decided to just start a band called the Lieutenant Dan Band, with the proceeds from the band's performances going to charity. He also stars in a series of ads as Lieutenant Dan supporting suicide hotlines for veterans. This leads to:
Tom Hanks based Forrest's accent on the child actor who played young Forrest (not Forrest Jr.).
When Bubba tells him "My given name is Benjamin Buford Blue, but people call me 'Bubba'", of course Tom Hanks would respond with "My name is Forrest Gump. People call me 'Forrest Gump'". This wasn't in the script.
Mykelti Williamson later appeared in season 3 of CSI NY as Chief Sinclair, the boss of Gary Sinise's character Mac Taylor.
Interestingly, Sinise also played George Wallace, whom Forrest interacts with in the movie, in a made-for-TV film.
Magician Ricky Jay designed a special wheelchair for Gary Sinise that used an illusion to hide his legs, which were on a hidden platform underneath. The contortion required to sit in it meant that Sinise could only be in it for about ten minutes at a time.