Hello!
I've been to Rome now several times.
Try going to a hostel. They are cheaper than hotels. But of course
there are less in terms of services. It might have a more younger crowd though.
The Des Artistes Rome Hostel has some decent reviews and it is near a main road.
My family rented an apartment when we visited there. It was nice, but you
had to walk a bit to get to the train station. Here it is, with pictures:
http://www.110apartments.com/new/en/property/view/rome/apartment/430
if you don't like that I think there are others in the list here:
http://www.110apartments.com/new/en/search/italy/?_o=town
Have a nice trip!