Just my 2¢ -

Since you'll be visiting the Colosseo/Forum/Palatine but also be heading into the heart of the Centro Storico, my thought for you is to be located on Monti's south western edge which is marked by the street Via Tor de' Conti, so you'll be somewhat in the center of where you're intending to visit.

Along the south/west side of this street you'll see an ancient wall, which was put up when Augustus built his forum on the other side of the wall. For anyone who's interested here's an informative page with an illustration of the Temple of Mars Ultor and its "Precinct Wall".

Here's google's streetview of Via Tor de Conti to "walk" down, with the "Precinct Wall" on view: https://goo.gl/maps/AqnSHt4tVj22

And here's google's map of Via Tor de' Conti, where you can see the hotels in that area: https://goo.gl/maps/gLRZtEFGmcr

These cover the gamut from the luxe Inn at the Roman Forum on nearby Via Degli Ibernesi to the homey, but well regarded Residenze Maritti. But in terms of receiving the best overall service with consistently well maintained rooms, I'd recommend the Nerva Hotel. Here's its official website.

They did a huge renovation years ago and raised their rates considerably from what they used to be, but I still consider this place a good value. But this is definitely one of those hotels where I see varying rates depending if the stay is towards the beginning of November versus the middle of the month, and rates differ from weekdays to weekends, etc.

So depending on your dates your budget may get you their 17sqm Boutique Budget Queen Room or, your same budget could get you their 39sqm Premium Suite. A few things to keep in mind if interested in this place: a room with a view affords you a view of that wall, wiith accompanying street noise (from the cars whizzing by, not usually from loud party people); their breakfast is nicely reviewed but it will cost you an additional €10 per person so it may or may not be a good deal for you; when booking you can also arrange for an airport transfer at €50, so if you were planning on taking a cab to FCO, using their service might prove easier. At booking they also offer tours - which I would not recommend arranging through them. You can also buy a Roma Pass from them, which is a whole other topic on whether it's a good deal in general, but given CHSBoater 's particulars, I would not think it would be a good value for them. One last thing to note: they offer a 15% discount if you provide a credit card to make the reservation, but pay in cash "during your stay". This kind of thing is usually a red flag to me - are they doing this for some underhanded tax benefit - but I've been reading about this place for years and it consistently earns high praise so I'm not concerned. For whatever that's worth!

You definitely have a ton more options than this one suggestion, but thought I'd throw it out there now because I'm going to be away from the internet for awhile, and I do rank this place at the top.

Happy planning!!