Since The Little Mermaid is not a fairy tale from the oral tradition, this page doesn't offer alternate versions of the tale from various cultures. However, Hans Christian Andersen did not exist in a vacuum and was certainly influenced by the popularity of mermaid tales and other stories of his time when he wrote his tale. Below I have provided bibliographic information and links to other stories and books which come from the mermaid tradition or are thought to have influenced Andersen. We start with the one which assuredly influenced Andersen's tale the most, Undine.
About 50 years after the Little Mermaid was published, Oscar Wilde published his own mermaid story, The Fisherman and His Soul. In this tale, a fisherman falls in love with a mermaid and seeks to give up his soul so she will accept him as a lover. Wilde's story incorporates mermaid lore and also appears to be somewhat of a reaction to Andersen's tale.