This resource has an explaination of the behavior, which relies on the salt-dissolved water wicking into the string. This would mean that you need a type of string which will absorb water. It's surprising to me that your nylon string doesn't work (as long as it's thick/low-density enough to absorb water, probably unlike fishing line), is it possible it has a hydrophobic coating?
I would recommend any cotton-based string or twine. Cooking twine and sewing thread is often cotton, if you have access to that. I've had no problems with these.
Edit: I just thought, that you might want to try this not in a cup of water first (like in the link above). You might potentially run into not-string-related problems if your water isn't cold enough, or if your ice gets too submerged when you put the salt on it. It's possible your nylon string is fine.