Does anybody know if one could stay at any of hotels near Sydney airport for a few hours during the day (7am to noon)? I know that such a service is available in many Asian countries but I cannot find any information on Sydney hotels.

I've never heard of it either, but these hotels are all large multinational chains, so I am sure you can do anything you want to do there, as long as you pay for it. I'm sure, however, there must be a better way to spend those five hours; how long before the 7 a.m. arrival, will you have been on a flight without sleep?

We are going from Chicago to LA and then to Sydney. I don't remember how many hours we will be in route but I am sure that it will be enough to want a normal bed for a few hours

Unlike many Asian and some European airports, there are no day-stay rooms at Sydney airport.

Some of the near-airport hotels may offer this - I see the Ibis is charging $115/night - this may translate to the same for a day stay. This is roughly what it cost me at Narita for a day-stay, a year or so ago.

Make sure you have a good alarm!

Don't go too far out, as Sydney can get traffic problems. Plus you are going to need to allow a fair bit of time for customs etc.
If you have a domestic connection within Australia the be warned that it will be in a different terminal to the international and the bus ride in between can take some time.
Good luck.

You should be able to get a day rate at just about any airport motel/hotel. It's just that most don't advertise or allow you to make reservations for day room. So basically once you get through immigration/customs at SYD, go to tourist help desk. They will be happy to provide some phone numbers or maybe even call the nearby places and inquire about day room availibilty. If the hotel was not booked fully the night before and is not booked solid for the day, they will offer you a rate. Usually it's the least expensive nightly rate. From experience the day room stay is from 8am till 4pm.

If you can't get a decent day rate, and you're really just looking for a bed and a shower, there is a Formule 1 hotel just opened earlier this year. They claim to be only 5 mins walk from the domestic terminal, 10 mins from international, and also have a free shuttle bus. Rooms are from AUD$69 per night. http://www.formule1.com.au/detail.asp?hotdeal_id=2007