Stay overnight inside the park. The website to book lodging is:
http://www.grandcanyonlodges.com/

That is the official website for lodging (it is run by Xanterra, the official concessionaire).

> Can you tell me where to stay for one night

If at all possible, stay within the National Park. ANY of the lodging is better than anything outside. The reason is that seeing The Canyon as the sun rises is one of the most memorable views on this planet, and getting to the rim before daybreak is difficult unless you stay inside.
Lodging is usually booked up month in advance, so even if you go to
http://www.grandcanyonlodges.com/
today, you may find nothing available.

If you find nothing, at least stay as close to the Canyon as possible. The key is to get as close to the rim as early as possible.

Do not be surprised if there is some snow there. The elevation is over 6000 feet on the South Rim. http://www.grand.canyon.national-park.com/weather.htm
Entrance fees are not refundable if you get snowed in.

There is availability in the park at all of the properties save two. Any preferences? Is one better located than another? Easier to get to from Phoenix?

el Tovar Hotel is the nicest property there, but Kachina, Thunderbird and Bright Angel Lodge are also on the rim. Some of the rooms in Bright Angel share bathrooms. Thunderbird and Kachina are basicly motels , but in a beautiful place.

They are all fairly close to each other so no real difference in terms of getting there from Phoenix.

Which ones are available?

rbnwdln, we stayed at El Tovar for a night in October. It is the most expensive of the lodges but a beautiful place to stay. If you want to eat in the restaurant you must book now. You dont have to be staying there but if you dont have a booking you wont get in. They have a nice little bar to relax and have a drink before dinner and their breakfasts are just a treat after getting up early to watch the sun come up on the canyon.(no reservations are taken for breakfast) Its a stunning walk along the rim near the lodges and staying overnight means you see it at sunset and sunrise. If you can get a booking stay on the rim. The canyon is just a few metres walk away rather than having to get in a car to see the view. You can walk outside your lodge at anytime to see the canyon and to me that was worth paying extra. Watching a big male elk walk through the lodge grounds as we ate breakfast was priceless.
Go to the website mentioned (Xanterra) and only book through them. There are maps of all the lodges so you can compare their sites and prices and availability.

If you can snag a cabin at Bright Angel...do it!!! If nothing is available now, keep trying as cancellations are not unusual since most people booked a year ahead and might have had a change of plans.

Bright Angel cabins can be charming.

If the Bright Angel Cabins are full - the Bright Angel Lodge is fine - especially if you get a rim view room.