Wiktionary or Etymonline are usually good sites for finding origins of words, but not this time. So I did a search for the term "flip-flop" on genealogybank.com and found references in newspapers as far back as the 1750s. However, the meanings weren't easy to get from context, but didn't seem to mean the same as today. In the early 1800's it seemed to mean shoddy. In 1866 there was an article where an editor of a newspaper was called the "ultimate flip-flop" who somersaults his way through various sides in politics. So this is the first time I saw if used in the same sense we use it today, with a reference to somersaults, where someone is actually flipping. I found a dictionary from 1857 that described flip-flop as "a succession of droppings, first with a sharp sound, then with a dull 'flop'". There was also an entry for flip-flap, "to flap backwards and forwards", which is actually closer to today's meaning of flip-flop, and further research shows that as far back as the late 1700's - flip-flap was a term for somersaults. Based on this little bit of research, I think the word evolved from Flip-Flap. Note that it was (and still is to some degree) very common to reduplicate a word and change a letter or two to create a new word. Examples include, hickory-dickory, hodge-podge, flim-flam, sing-song, Flip-flop probably began this way.

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Politicians usually get called flip floppers because they change their opinions all of a sudden.