What does open jaw mean.

Open jaw means you fly from one city into another and then return via a different city (example: fly from your home city to London, tour Europe and fly home from Rome, your last European destination). On booking sites it's the "multi-city" option.

Flying open jaw (also known as multi-city or multi-destination) in your case would mean flying into London and flying home from Rome. It is usually the same price or not much more than a round trip ticket like you purchased. You can save time and money flying open jaw by not having to pay to return to your entry city and wasting a travel day. How do you plan to travel from Paris to Rome?

I can't advise on intraeuropean flights, but I'm sure others will come along.

$123 to me is cheap enough. You could check whichbudget.com to find budget airlines, and you likely will find cheaper advertised rates, but make sure you consider all fees and what might be added for baggage. Do not assume you will carry-on with a budget, usually you are limited to one item, small, weighing less than 22 lbs, and they do check to see if your bag is over. British Air may be a little more liberal, but plan on checking a bag.

The only other caution I have is do not try to cut your flight from Rome and your Flight home too close. Even booking them for the same day could be too close since any delay on your Rome-London flight may mean missing your flight home. Ideally fly to London the day before your flight home to be very safe.

Yes there are cheaper flights from Rome to London although the budget airlines should be priced well in advance to get the best deals. You can look at www.skyscanner.com as well as Whichbudget to find flights that don't come up on other search sites.
BUT: Beware of a couple of things. easyJet, for instance, a fine carrier, offers a one-way on that route for under $70, if bought long in advance. However there are extra fees that they do not display on the opening page, unlike the trans-Atlantic airlines which now must tell customers at the start what the total is likely to be, thanks to American rules. Despite those fees you still could save some cash.
Here's the other BUT: Most of the no-frills carriers offer no connections to other airlines.
They don't even connect to their own flights. Each segment is sold separately. So trying to fly back to London to catch your homeward-bound plane the same day carries significant risk. You need considerable time to retrieve your luggage, check back in with the next flight, and clear security.
That BA flight, however, might still allow you to check your luggage all the way through, but it would have been much easier to arrange if you had bought both flights on the same itinerary ticket.
Sorry, but the clear advice for next time is to think open-jaws right from the start and then fit any supplementary tickets into that structure.
PS: (I'm obsessed with colons here) London-Paris can be a thrill by the tunnel train and competitive with airlines in both time and money if the tickets are bought well in advance. From Paris to Rome, though, I would fly, and did six months ago, on easyJet.

Easyjet's prices show every cost for a basic flight - no add-ons for taxes, admin fees etc. It costs more than the headline price only if you want to book extra services, eg reserved seat, checked bag(s) etc. Fairly clear.

The caution about different airports with the budget airlines should be a strong one. If you want to fly Rome to London and on back to the US the same day, make sure your flight from Rome comes in to the same airport as your departing flight ( probably Heathrow). Since EasyJet uses other airports ( not sure which) you would have to do a very stressful intra-airport transfer, or build in a night near London,,which adds expense. Even a BA flight to Heathrow carries some risk of a missed connection. You might want to put some of your London time at the end of the trip,to take care of this.

Had you booked open jaw originally (flying in to London and back from Rome), the flights would all be on one ticket and even if your flight from Rome was late and you missed the scheduled connection to the long-haul flight, they would put you on a substitute flight. With flights booked separately they have no obligation to do this, even if it is the same airline for both flights.

BA allows for free: 1 carryon, 51 lb/23kg, 22”x18”x10”, and 1 personal bag/brief/laptop, 18”x14”x8” must fit under the seat.

They have franchise carriers: Comair, Carryon limit 15 lb, 22”x14”x9”;
and SunAir, Carryon limit 13 lb, 18”x14”x8”.

On my inter-continent trip portion which was firts leg back to U.S., I could get the size but not the weight. Thus, I had to check my bag under. My bag weighed 22 going over and 27 coming back as I never mailed my gifts home.
For the second segment back to the U.S. I was okay but bag was already checked through.

Double check your plane from Rome to London.
Enjoy!