As with most weird concoctions in the world, the origin of mermaids, those semi-human, semi-fish creatures, goes way back into the depths of mythology. Their history is far older than the stories of early European sailors who mistook manatees (sea cows) for beautiful maidens (should have gone to…).
Mythology aside for once, a possible source of the idea can be found in the Far East. Since very ancient times, pearl divers were women because they could tolerate the cold much better than men (nothing changed there) and learned to hold their breath for several minutes. As they wore no clothes, they would pop up out of the water and hyperventilate to get some fresh air and warmth into the system. To encourage this they would sing very loudly to each other – hence, possibly, the 'siren song' associated with mermaids luring hapless sailors to their doom (unless that's from the ancient wife in the kitchen as her ancient husband comes in the back door with muddy sandals – same sort of noise level).
There were many human/animal beings thought to exist in the distant past. It could be a way of showing the direct and deep connection our ancestors had with the rest of the fauna on the planet – they saw themselves as very much within nature, not lording it over nature. For several thousand years this notion has been relegated to a minority view. But it is making a comeback, with groups of people demanding 'human' status and rights for our nearest cousins – chimps and gorillas. If granted it would raise the interesting question as to whether or not they also have souls…