Don't know about tiling, but we had good luck with cork flooring from home depot in our old basement. We put down a thick vapor barrier and some insulating underlay and then the snap together flooring on top. The cork was able to expand and contract because it was just clicked together, and we could pull out and replace a plank pretty easily if it buckled or got damaged. It made the floor a lot warmer. The vapor barrier stopped the little bit of moisture coming through.
Mind you I have no expertise in this and I'm not sure we did it right. But it was relatively cheap and easy, looked good and worked at least for a few years. We moved, so can't comment on its longevity.