Rome is a fairly noisy city but it's part of it's charm - latin and loud!
There is an offer on
Groupon at the moment and they are offering accommodation at The Black Hotel - which might be worth a Google if you can't see the link...
However this hotel doesn't get very good reviews with
Tripadvisor which is a site I'd rate when looking for accommodation :)
I'd recommend you decide what you MUST see in Rome and plan your location accordingly. It's all walkable if you're fit but you can easily find yourself half a city away without realising. Try and avoid budget hotels too close to the station though they are all pretty loud and people are checking in at all times which is a nuisance