The ice could freeze enemies in place. The ice sword would be unlocked immediately, but the ice arrow and ice hammer would be unlocked though the bow only challenge and the hammer only challenge. Ice breath would be unlocked though a challenge like the inferno challenge, but with ice based enemies. You would not be able to have a fire sword and an ice sword, once you got one the other would get a cross though it.

New ice traps would include ice platforms that you would slip on, and ice platforms that would freeze any part of you that touches it

Maybe when you get it by a frozen weapon that part of your body gets frozen, and you can't use that part for some time until you thaw out (eg. one of your legs frozen = can't move for 3 secs. one of your arms frozen = can't use the bow for a few seconds)

It will be cool if frozen robots parts are harder to cut through since there is an extra layer of ice to cut through, so a strategy would be to freeze their legs (so that they can't run around) or their arms (so that they cant attack or block with sword) and attack wherever they are vulnerable. Maybe the fire sword slices through ice like nothing.

There can also be an upgrade where you thaw out much quicker.

Its likely this game would have elemental weapons considering it has fire