Yes, that's currently the question facing the two primary producers of single-aisle aircraft, Airbus and Boeing, and one of them has provided an answer thus far. It's hardly a new idea, however, as a brief review of the history of re-engining of commercial jet aircraft shows.
There are two categories of re-engining to consider: New/different powerplants installed on existing aircraft and changes within ongoing aircraft programs. Both are typically driven by a variety of factors, including improved aircraft performance (such as better takeoff characteristics, or greater range); economic efficiency (reduced cost due to lower specific fuel consumption, decreased maintenance costs due to improved reliability/longevity, etc.); and environmental factors (including noise, and more recently, emissions).
In both cases, there need to be sufficient economic incentives, and/or more stringent regulatory requirements (such as stricter noise regulations) for a change to be considered, whether for existing equipment, or a change to an ongoing program. In the former case, there also will be concerns about the downtime for modifying the aircraft, when it will not be producing revenue, in addition to the direct costs associated with the modification; in the latter, the large-scale cost implications of significant expense will need to be weighed versus factors such as the future competitiveness of the type if it is not re-engined, and the degree of improved performance and operating economics that will be achieved via taking this route.
In the turbine power era, there have been several significant re-engining programs. One of the first involved installing turboprop engines on Convair 240/340/440s that were built as piston-powered aircraft. This involved three different manufacturer's engines, including Napier Eland (Convair 540); the Rolls-Royce Dart (Convair 600/640); and the Allison 501 (Convair 580). This proved to be an excellent way of keeping what proved to be very sturdy airframes in service for much longer than would have been the case had they remained piston-powered.
Early in the jet age, American Airlines and Qantas both re-engined early 707-100 aircraft, to obtain better performance. This involved replacing one type of Pratt & Whitney jet engine with another, going from the JT3C to the JT3D. This sounds like a relatively minor change, but it converted the aircraft from turbojet propulsion to the far more desirable turbofan configuration.
The 707's contemporary, the Douglas (later McDonnell Douglas) DC-8 also had a significant conversion program in its history. Following the advent of more stringent noise regulations in the 1970s, the JT3D now proved to be a liability. Since the DC-8 airframes in the 'stretched' category (-61 and -63) were fairly recent, provided good economics from an airframe standpoint, and had excellent conversion potential as freighters, a number were converted to use CFM56 engines produced by the partnership of GE and Snecma in the 1980s.
On a smaller scale, there was another conversion of cargo aircraft, for a single customer, United Parcel Service. UPS elected to replace Pratt & Whitney JT8Ds with Rolls-Royce Tay engines on a portion of its 727-100 fleet, to meet more stringent noise requirements. In addition, the parcel carrier also made significant upgrades to the cockpits of these aircraft a part of this program.
While it was once common to have only a single engine choice on an aircraft program (although even the early 707s and DC-8s offered both Pratt and Rolls-Royce engines initially), today many programs offer at least two choices, and there have been instances of the so-called "Three on a Wing," where engines from all three major engine manufacturers (GE, Pratt and Rolls) have been available.
On the subject of engine changes within programs, the 737 is the most prominent example, with Boeing launching the now 'Classic' series, beginning with the 737-300, by moving from JT8Ds to a variant of the CFM56 engines utilized in the DC-8 re-engining. Interestingly, even the CFM56 has now been supplanted on a type, when Airbus chose to replace this powerplant, used on the A340-200/300, with Rolls-Royce Trents for the larger -500/600 models. McDonnell-Douglas utilized a higher-bypass version of the JT8D to morph the DC-9 into the MD-80; further development, into the MD-90, involved moving to the IAE V2500 engine.
Now, Airbus has decided to upgrade the A320 program with what it calls the Neo (New Engine Option). The CFM product remains on offer, via the "Leap-X" version of the now-veteran engine; interestingly, Pratt returns to the single-aisle field with its PurePower PW1000G geared turbofan, while IAE's V2500 is dropped.
And thus, it's time to wait and see what Boeing will do with the 737. Will the fourth time be the charm in this case...or will they begin anew, as they appear to be leaning today?