What does gin mean like in cotton gin?

There is no definitive answer in the research. However, it might be that the word gin in the name is a reference to the word Djin, a genie from the Arab world. It is known that the Victorian era used such words in their advertising, so it is a possibility that the word was a working name for the seed extractor. Being American and having a propensity for shortening names and words, the original Djin probably became gin over the time of the development of the cotton gin.