I think you're still right that it comes out cheaper, though, because you'd be spending most of the freezer's energy output anyways.
This is also why it is a good idea to de-ice your freezer every now and then, keeping all that ice at a low temperature against the leakage through the walls of the freezer costs energy.
I've always been under the impression that though you pay a penalty for freezing things the first time you put them in, thereafter a full refrigerator (and freezer) is actually much more efficient because it's more difficult to keep air cold than it is to keep any variety of items that are more dense. If we're talking about leakage, too, air obviously exits the freezer much more often than its contents do.
If you imagine a giant freezer filled with ice it will eventually melt. This takes energy, a larger freezer will take more, an empty one will take less than a full one.
Counteracting the heat transferred through the freezer walls from the outside in will require an amount of energy proportional to the mass in the freezer and the leakage from the outside.
It takes energy to maintain that imbalance, the bigger the imbalance the faster the heat will transfer.
If the stuff in the freezer is light (air) then it will take less energy to keep it cold than if the stuff in the freezer is heavy (ice), because there simply is less leakage from air (a thermal insulator) to the walls than there is from ice (a pretty good thermal conductor).
Thermal conductivity is not very tightly coupled to mass but it usually is a good indicator, in the case of ice vs air it works pretty good.
So, it takes more energy to keep a freezer full of ice at a low temperature than a freezer full of air...
For an encore, which is heavier: A cubic meter of dry air vs a cubic meter of humid air ?
In terms of keeping it cool, that probably depends on how good your freezer's insulation is. If it's really good, it probably doesn't take too much energy to keep things cool (air or otherwise). If it's not so good, you'll incur a constant "your freezer sucks" tax.
In the case being discussed, we're talking about freezing stuff every day, and letting it thaw in/near your bed during the night. So the cost would definitely be there.
The problem is so under-specified at this point it's kind of silly to pretend we could get a meaningful answer yet. Are we talking about the ice/fan and a/c cooling the room to the same temperature? Or are we just talking about the sleeper's comfort, in which case we should take into account that i/f can be put right next the sleeper while an a/c has to sit all the way over in the window? Etc, etc.
I have another question, since my basic physics knowledge is a bit lacking. Compressed refrigerant system like air conditioners work by expending (electrical) energy to move heat from one system (the inside air) to another system (generally outdoors). With the ice-fan, where does the heat go? This is probably a fallacious way to explain it, but is the energy being removed from the air and used to transfer the state of the H20 from solid to liquid? And if that's the case, if you wanted to cool an entire room rather than a column of blown air, you'd have to have the liquid water end up outside the room, just like the exhaust from an AC system is outside?
