Lots of good info already in here. Stretching the line gently and/or using the hot water soak to remove memory; using large spools like a 2500-sized summer reel on an ice rod; using the softest flourocarbon line possible like Seaguar Tatsu or Invisx.
I’ll add another: line conditioner. I’m a huge fan of KVD Line and Lure Conditioner for conditioning line and making it softer. It also tends to make the line last longer before it loses that shiny finish and starts to get chalky-looking. It will make braid keep its color longer, too. It’s made by the same company that produced Bow to Stern (BTS) fiberglass boat conditioner, a spray on product that will keep a bass boat looking freshly waxed all year.
All in all, I don’t use fluorocarbon on ice fishing reels. Too much hassle. Limited benefit. Plus the softer fluoro stretches worse than monofilament, not that it really matters 20′ under the ice.
I use either braid with a fluoro leader or my favorite open-water line for moving bass baits, like spinnerbaits or crankbaits. Check out Yo Zuri Hybrid (not Ultrasoft. . . junk). It’s the lowest stretch, most abrasion resistant, strongest nylon line on the market, as far as I’m concerned, and it’s relatively cheap. Google “Yo Zuri Hybrid Breaking Strength” for some more information. In stretch tests, it always ranks right at the top. I use 4-6lb on my walleye rods, most often 4lb. The 4lb line has a diameter of most company’s 6-8lb line, handles like a 6-8lb mono, and breaks at about 12lbs. Sometimes I put a swivel and a leader of fluorocarbon on it.