Very much weather dependent- snowshoeing, cross country skiing, possibly ice skating. It might be too warm for those things, tbd.

Art museums - MIA or Weisman are free

Historic tours - James J Hill house, Swedish Institute, Mill City Museum

Arcades - Can Can Wonderland, Uptown (21+), Tilt Pinball

If it's nice enough to be outside - walk across the Stone Arch Bridge and also stop in the Guthrie to get a view off their balcony

Random - US Bank stadium tour, spam museum in Austin (requires a car), there's a variety of comedy shows, Hennepin Theater Trust, Midtown global market, Paint night (cheers pablo), axe throwing, indoor rock climbing. Kind of depends what you're interested in.

Check citypages for events.