I would stay anywhere but the termini station area. There are some real seedy areas around there.

Staying near the Pantheon is a great area, I agree with the others. It's central, walking distance to nice family-run ristorantes, and to the Trevi fountain, Piazza Navonna. Take public bus or taxi to the Vatican Museums and St. Peter's. Consider walking to the Colosseum and then taking the public bus back.

We loved the Hotel Nazionale a Montecitorio on Piazza Montecitorio, near the Pantheon, but the price of rooms was a little more than you want.

The Pantheon area is the best place to stay. We spent 4 nights in Rome in April and thanks to the location of Hotel Torre Argentina, we were able to visit the most important places by a short walk.
We got a special price from www.reservationitaly.com.
We paid Euro 153,00 for a double room per night.
One more thing. Pizzeria Panattoni in Viale Trastevere. The best pizza I have ever tasted!!!

We also loved the Hotel Nazionale Montecitorio and its location, as Melissa said. We got a great deal on the internet, and it's worth looking for. We stayed a week and walked absolutely everywhere in the city...I think in an entire week we only got in a cab one time.

Check out www.ReservationItaly.com I got an excellent price,the best yet of the 3 times we have stayed at our top choice hotel, Ponte Sisto. It is well located, just steps from Campo dei Fiori. The site has many hotels from which to choose, so the suggestion to also check with Trip Advisor is a good one.
Carol L