Personally, I'd recommend caging her for the winter so you can keep her warm enough. Otherwise, you're going to need to keep your entire room at 75 degrees or so to make sure she stays warm enough, and that may not even do the trick - floors are usually draftier, and there's always some spots that aren't as warm. If she ends up someplace cooler, she'll likely attempt hibernation. And as you said, if your space heater goes out and the room temp drops, that will probably cause one as well. Your mom is (no offense to her) not right - even if you put down a heating pad or snugglesafe for her, if she does stay by it, that means she won't exercise, eat, drink, etc. If she doesn't stay by it, but goes back and forth, the change in temperature can cause a hibernation attempt as well. The air wherever she is needs to be warm, not just create a warm spot for her to retreat to - they won't always do that, and if they do that, they may not leave and exercise as they should.
For a big enough cage that's relatively cheap, you can build a C&C cage, but those can also be hard to heat - you'd probably need two CHE lamps and you'd need to insulate the cage, unless you're planning to keep your room at 70 or above as well. Here's a sticky with more info on heating options - viewtopic.php?f=8&t=4579
Power outages are definitely a risk, and there's a sticky from Nancy on that - viewtopic.php?f=8&t=325