I would go with the braid or any of the superlines. Typical memory is less and can last in my experience up to 5 years easily. You usually don't use much line especially ice fishing. Stretch on superlines is about 0 and mono especially the limp stuff can be up to 25%.
XL will be cheaper, but much more line memory and if you are lucky will last maybe a year. If you do use XL or any mono, especially on small reels, be sure to put the spool of line on the floor before spooling. If the line twists while spooling onto your reel (spinning) flip the spool over and continue spooling and it should go on smooth.
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