Your best bet for a cheap budget hotel in Rome Italy is with a programme/company called HotelsCombined.
They will search through many other Rome hotel bookers like booking dot com, venere.it, orbitz, priceline etc. and then take the best rates for a particular hotel(s) from all of them. They have thousands of hotels in their data bases.

You can visit hotels in central Rome for more info about what to consider/check when booking a hotel in Rome.

and then visit central Rome hotels to search through around a thousand hotels in Rome! (When you finish your search you can click on "price" and it will return Rome hotel prices from the cheapest - sometimes you can find some excellent deals.)

You can also have a look to see which area of Rome will most suit your requirements (what you would like to see, do, which Rome tourist attractions you want to be near to etc.) and where Rome public transport is, and how to use it etc.

...Oh yes, and be sure to check the Rome hotel reviews section for a hotel that you may be interested in. These comments are submitted by guests that have stayed in that particular hotel and are usually a very "cut to the chase" indication of what to expect from that Rome hotel.
Enjoy!

Rome Italy hotels guide