The question may seem silly but I have had the same problem myself sometimes. There are different sollutions on this problem, depending on the circumstances. In my discussion below I use two fictive caves; Nasty Cave and Muddy Cave.
For your info; here in Sweden even very small caves have different names, depending on local history and who you are talking to. In the swedish cave catalogue all different names are listed for each cave, but of course each cave does just have ONE ID-number in our database.
I hope that there will be more comments about this question!
A Double Barrelled name I guess. (eg Peak/Speedwell or Mammoth/Flint Ridge)
In a nut shell, whatever name you want to give it! Of course everyone might not go along with your choice. For instance if a relatively insignificant cave connnected to Mammoth or Carlsbad, I doubt anyone would go along with a new name for the "system". But even when major caves conect there is no set convention for the resulting system name. Sometimes the larger or more well-known cave simply changes its name from "cave" (or whatever) to "system". An example is that people commonly refer to the Mammoth Cave System, in spite of the fact that several significant caves have been connected together. Another common scheme is to use both names and hyphenate the result. Flint-Mammoth is another commonly used name for the Mammoth Cave System, because this highlights the first major connection that was made. Of course this name totally leaves out other, later significant contributions to the system, such as Toohey Ridge (Roppel).
Perhaps best, although potentially more confusing, is to make up an entirely new name. Sistema Huautla (Sotano de San Agustin and Li Nita initially, later joined by La Grieta and Nita Nanta) and Sistema Purificacion (Cueva del Infiernillo and Cueva del Brinco) are examples from Mexico. Generally the new name is based on some geographic feature or area. This method avoids leaving out any later major contributors that may be connected, and it avoids the feeling that one cave has "taken over" or ecclipsed another. It also avoids using an overly-long, multiply-hyphenated name.
It could be "A", it could be "B", it could be a portmanteau name like "AB" or "AB-system", or it could be "C". There are no rules. Sounds trite? Well, there are plenty of examples of all of the above.
If one cave dominates the other, then the combined system is likely to get the name of the dominant one (eg. Gaping Gill has remained the name for an increasingly long network of caves now with five or six other entrances, Raucherkarhohle is still the name for a cave which has at least 21 other entrances linked to it).
Very often the name which is used is one which has existed to refer to various caves collectively even before they were connected, such as the name of the hydrological system of which they form part (the Riviere St. Vincent is the name for the Gouffre de Pierre St. Martin and all its feeders, the Easegill system is the name used for all those caves draining to the underground course of water from Easegill Beck in Yorkshire) or the ridge or hill under which the caves lie (eg. Flint Ridge (since connected into an even larger system with a portmanteau name); the West Kingsdale system includes many sink caves and two valley entrances), or some feature which distinguishes this collection of caves from others (eg. if you include connected (and as-yet unconnected) caves further afield from the Easegill and West Kingsdale systems, then it is the Three Counties System). If all the caves in an area start to connect together, then the system may take on the name of the area itself (eg. the Huautla system).
The Italians typically name any significant cave after a person, and a "new" cave made by joining up ones with existing "geographical" names is viewed as a particularly good excuse. The germanic nations particularly go for long strung-together names (eg. the Platteneckeishohle-Bergerhohlensystem) because their language works that way, but once you get to Stellerweg-Schnellzug-Larchenhohle-Schwarzmooskogeleishohlensystem, then the people maintaining World Long and Deep lists either complain that your cave is ruining their layout, or just abbreviate it anyway :-). If various different caves were explored by different groups (as in this case), then no one group may feel they have the right to choose a name for the combined system, especially if those groups are not in contact with each other and are all foreigners in the country ! Binary names are particularly popular if the two caves linked are a sink and resurgence, as the name emphasises the through-trip nature of the cave (eg. Calf Holes-Browgill Cave).
It is probably most often the case that the name evolves from usage, rather than being deliberately chosen. In countries where all geographical names go through some sort of approval system, then an "official" name may be bestowed (eventually), but that doesn't mean that it is the name that cavers will use.
I expect that completely fails to answer your question, but then cavers generally don't like rules of any sort :-).
In Alabama (USA), we have an established guideline for the result of connecting two caves. The name and index number of the cave that was found first - i.e. the more established name in most cases - is adopted for the new system. Often the other cave retains its name for the entrance only, as in "The BrandNewCave Entrance of OldEstablishedCave".
I can imagine some cases where one would want to ignore this and create a composite name, and that has been done in the past. When Gross and Skeleton caves were connected (before my time!) the amusing result was Gross-Skeleton Cave, a good place for a Halloween trip.
It is important to be able to refer to parts of a complex cave by a name, which facilitates communication and makes the system easier to learn. I think it preferable to have a separate name for each entrance, rather than having to resort to a tedious description of how to get there. However, place names evolve pretty naturally in most cases; I don't think strict rules are necesary.