Not a hotel, but my family of 5 saved a lot of money by staying at an Air BNB in Piazza Mazzini. It was a large 3 bed, 3 bath apartment which allowed us to spread out and each have our own space at the end of the day.
It was nice to be out of a touristy area and be in a Roman "Neighborhood". Obviously no room service or cleaning, but there were quite a few small restaurants, coffee shops, and grocery stores in the Piazza and a produce stand that would set up shop each morning. Different strokes for different folks, but we really enjoyed it.
It was about a 10-minute walk to the Vatican and a block or so away from a Metro stop, so it was easy to get around.