The story
According to Michael Quinion of World Wide Words, "It has plausibly been suggested it came from the use of mixtures of cotton and other fibres in clothing. In the early 1800s, to cotton to somebody implied that you were drawn or attached to that person. It may be that the idea here is how well a thread of cotton sticks to the surface of cloth."
Etymology from Wiktionary, CC BY-SA 4.0.