Ah, finally something in /r/askhistorians I can chime in on. I'll just list all the blatant historical nods located within the film...

-The name Forrest come from Nathan Bedford Forrest who was a confederate general and the first Grand Dragon of the KKK

-Elvis Presley stays at his house when he is young and apparently Forrest, with his awkward dancing due to his legs in braces, is the inspiration for Elvis' dancing style.

-Goes to football at Alabama where the coach is the legendary Paul "Bear" Bryant who is known for both being one of the greatest college football coaches of all time as well as wearing his signature houndstooth hat.

-He becomes a college all american and meets President Kennedy. Forrest mentions how he was shot later in Dallas.

-Being at Alabama means he was also present for its desegregation and the famous stand at the schoolhouse door by George Wallace. Forrest helps Vivian Malone with her dropped book as she first enters the school. Forrest later mentions how Wallace was shot while running for president.

-When Forrest joins the Army, he sees Jenny in a Playboy which was right around the time of the height of Playboy's popularity.

-Forrest is with the 47th Infantry Regiment, 9th Infantry Division in Vietnam. That unit was in Vietnam from 1966-1971.

-Forrest gets the Medal of Honor and meets Lyndon B. Johnson.

-When Forrest gives his muted speech at the anti-war rally, he is introduced by Abbie Hoffman who was a major political and social activist that started the Youth International Party and was a member of the famed Chicago 7.

-Jenny's boyfriend during this time is the president at SDS (Students for a Democratic Society) at Berkeley and Forrest goes to a Black Panther Party get together. Both the SDS and the Black Panther Party were major counter-culture political and social groups in the late 60s.

-The ping pong section of the movie is a throwback to Ping-pong diplomacy in the early 1970s with China where the US and China exchanged ping pong players which led to better relations between the two countries and eventually leading to Nixon visiting China. This is why Forrest is invited to the White House again, is put up in the Watergate Hotel by Nixon, and evidently leads to the discovery of the Watergate scandal when Forrest calls Frank Wills when he saw people with flashlights from his room window.

-Forrest goes on The Dick Cavett Show for his ping pong exploits and meets John Lennon. Forrest, to the delight of a curious Lennon, describes China with what would later become lyrics for Lennon's most famous solo song "Imagine."

-The storm that Forrest and LT Dan are caught in is Hurricane Carmen.

-Forrest makes the bulk of his money by investing in Apple Computers. Although this doesn't follow history that well because Forrest gets the letter saying he doesn't have to worry about money anymore in 1976 and Apple didn't go public and make it's monetary supporters rich until 1980.

-Forrest running across the US coincided with a major jogging craze in 1978-1979. During this time he influences the popular bumper sticker "shit happens" and the smiley logo with the phrase "have a nice day". Although both of these are also historically inaccurate since both existed or were popular prior to 1978.

-As far as the story being inspired by an actual person running across the country, Stephen Faust walked across the country in 79-80 and was popularized by doing a live weekly radio show at the different locations he visited. Peter Jenkins walked across the country from 1973 to 1979 and wrote about it in National Geographic and in his book "A Walk Across America." Dale James Outhouse walked the US in 1984 and wrote the book "Walk of the People - A Pilgrimage for Life." As far as a guy running it in the 70s or 80s and doing it because "they felt like running" it doesn't seem to be based on any specific individual.

-In the background, the assassination attempt of Ronald Reagan can be seen on a TV

-Jenny is assumed to have HIV, which obviously was a big scary deal in the 80s.

Most of the events in Forrest Gump are at least loosely related to real events that took place. A person who grew up in the 60's and 70's would likely recognize most of the events in the movie as they lived through them and saw them on TV or read about them in the paper. I don't think there was anything particularly obscure or hidden, but depending on your knowledge level you could easily miss a lot of the nods to history.

Other examples don't necessarily cover what you might think of strictly as "history." For example, someone who doesn't follow college football might not recognize Paul William "Bear" Bryant, Forrest's head coach at the University of Alabama and arguably one of the greatest coaches in the history of college football.

The most subtle nod in my opinion is the interview of Forrest on the Dick Cavett show along side John Lennon. Some of the words of the interview are lifted almost word for word from the song "Imagine" and it is meant to imply that Forrest inspired one of Lennon's most famous and influential songs.