In reply to ablackett:
You're not mad! We went in January last year and drove right round the island over two weeks, and despite the heaviest snowfalls they'd had for many years (apparently) we had a really amazing adventure and saw plenty of what is a really unique and amazing place.
Check out sadcars.is for rental, when we went it was far and away the cheapest place. I would definitely recommend hiring the biggest 4x4 you can afford, as driving conditions will be hairy even if there is not as much snow as last year. We had the cheapest 4x4 which was actually a car and though it did very well, I'd definitely upgrade if I did it again.
Myvatn in six hours is probably optimistic - doable in good conditions but the snow slows you down a lot, and don't underestimate how tiring it is to drive in! Requires constant attention. Eight hours there, or doing it with a stopover is probably more realisitic. Well worth the visit though, the nature baths there are in an incredible setting, and will be pretty quiet if not deserted during the day.
Route 1 is generally pretty good. We often found it covered in deep fresh snow, but that is manageable for driving in - it's the ice that's really a problem and the only place we encountered that on route 1 was in the north-east corner on the way to egilstadir where the locals don't really use it and they don't bother opening it much, so bear that in mind if you plan to go further than myvatn.
If you do, Dimmu Borgir is well worth a visit (beware the icy approach road!) especially at that time of year, when the Yule Lads will be out and about getting up to mischief. We were too late for them sadly!
Daylight will be an issue too, but is manageable if you plan so that you arrive at your destination at sunrise. We had six or seven hours which was fine, but you'll need to plan really well with only four. It's the perfect time of year to see the northern lights though, we managed to see them a few times.
We didn't hike, as we didn't have appropriate gear, but we did venture on to the very bottom of a couple of glaciers. Based on the amount of ice around generally I'd say crampons would be a prerequisite, and snow shoes would be wise. You'll also need to plan that very carefully as you'll likely not find hiking maps until you're there (bookshops in Reykjavik are a good place to get them) and just getting to the trails could take a long time.
As for places to stay, you're pretty limited to what's available - as you say the hostels and campsites will all be shut. You'll find places in the bigger conurbations without too much trouble, but for the more remote areas Icelandic Farm Holidays (www.farmholidays.is a partnership of farms with varying degrees of accomodation) is invaluable, if a little expensive for what you get.
Right, stream of consciousness over, I could go on and on but I'll have a think of any other useful tips. Obviously all this is only my experience but if you've any more questions I'll happily share as much of it as I can!
If you plan it right and know what to expect it really will be an amazing trip and I wish you all the best.
Ben