"Little Egypt" is the name of the belly dancer who, legend has it, drew people to the fair (and scandalized the arbiters of morality) with her "hoochie-coochie" dance (although there are no records from that time that indicate anyone performed at the fair under that name). After the fair, several dancers claimed to be THE Little Egypt, and the myth of her performing at the fair took hold.
Something that did originate at the 1893 fair was a melody improvised by Sol Bloom, the manager of the Middle Eastern dancers at the fair (some of whom danced at an exhibition called "A Street in Cairo"), which was later appropriated by other composers. The most popular version of this song was called "Streets of Cairo or the Poor Little Country Maid." Although no one today remembers the lyrics, everyone knows the melody - it's the "da da DA da daaahh, da da da DA da da da" heard on cartoons, commercials, spoofs, and sung by kids with lyrics such as "All the girls in France do the hoochie-coochie dance . . ." You can check Shira's site here for a song clip and more history of the song (the song clip plays right away, so don't try it at work). A short write-up on the various Little Egypts can be found here.
So what does all this have to do with what's going on in Egypt today? Nothing, of course - but after all these years of studying bellydance, this kind of free association just can't be helped, I guess.