I am currently in Cairo. If you do a search, there is a long thread on Cairo which discusses which places to see.

Let's be clear, though. One day off means you'll get to see the pyramids and not much else. Firstly, because the traffic is terrible and secondly because the heat, even now in early June, is intense. After 2 and a half hours at the pyramids, I collapsed into the Oberoi Mena House (a lifeline, a saviour - there is no cafe of any sort in the pyramids) for an air-conditioned lunch. The round trip with lunch was about five hours from my hotel in Heliopolis. If you're staying in the centre, its still going to be 4.5 hours to do it properly, with lunch. You'll then want a shower and rest back at your hotel, I promise you.

Note that if you want to go in the Cheops Pyramid, which is amazing but you MUST be in good physical shape (a walk through a four foot high passageway that is very long and slopes at 45 degrees) then only 150 tickets are sold each morning and afternoon. My guide book said that tour groups tend to get the AM ones, so you should arrive around 1pm when they start selling the PM ones. You need to buy the Cheops ticket inside the site, not outside.

The other 'must do' is the Egyptian Museum which, again, will take at least 2 hours to do properly plus travel time there. Its open until 6.45pm so you could do it one evening.

Don't plan on doing much else given your small amount of free time. What I WOULD recommend is visiting the Four Seasons Nile Plaza, which is just superb and has wonderful dining. And when you fancy a burst of the civilised world, take a cab to Citystars in Heliopolis - a new 250-shop shopping mall / cinema / theme park which looks like it fell out of the UK, albeit that most of the shops are local brands.

Of the off-beat tourist spots, the Gayer-Andersen house is very interesting. It is also next to a huge mosque and a short walk from one of the other major mosques, which itself is underneath the citadel (although the entrance is on the other side). Hiring a car and driver for a 3 hours should get through this list. I had Gayer-Andersen and the mosque totally to myself yesterday afternoon. Even the pyramids were relatively tourist-free today.