According to TV Series Finale, due to a stipulation in Tina Louise's contract (she was Ginger, "the movie star"), she was to be the last person billed in the opening credits. The parts now known as "Mary Ann" and "The Professor" were re-cast to Dawn Wells and Russell Johnson respectively, but only after Louise's contract was signed, so they were placed in the end credits.
According to this Today Show interview (YouTube, @ 3m 30s), Bob Denver notes his contract stipulated he could choose his billing and for season two he declared he would put himself in the end credits unless Wells and Johnson were moved to the beginning. Either Louise agreed to change the credits, or her contract was reworked.
Another variant on the song took place during re-runs, and I witnessed it. Some syndicates re-running the show manipulated the second half of the theme song, the part which ran during the credits. The syndicates silenced the broadcast audio during the words "each week" so
"{...} so join us here each week, my friends {...}"
became
"{...} so join us here {SILENCE FOR TWO BEATS}, my friends {...}"
I presume this was done since the re-runs were broadcast on more frequent intervals than weekly, and the associated networks questioned the intelligence of the viewership.
Additionally, The Gilligan's Island fan site hosts an alternate theme song played during the pilot episode.
I know of no other variants.