It will depend on political facts
You have to ask yourself two questions:
- Why are these countries merging?
- How will the newly formed nation be organized politically?
The bottom line is that naming a country is not like naming a baby: people aren't going to simply sit around and toss out words and names they've "always liked," and then pick whatever seems like a "good fit."
Why are these countries merging?
If they're doing it because they all belong to the same religious tradition, the new name is almost certainly going to reflect that fact, probably referencing by name the sect itself or some character from its mythology, or some aptly-chosen episode or principle from its traditions. For example, if both countries are filled with people who worship the Flying Spaghetti Monster, the new name is probably going to mention pastafarianism or noodles and appendages. ("Pastafaria" is a likely choice.)
If they're merging because the monarchical families of both nations have been intermarrying for generations and they've decided to simply pool their political power and resources, the new name is almost certainly going to be a nod to the personal histories or pedigrees of those royals, or to some symbol of their unity that they already cherish. That might be something as straightforward as the name of the estate that is traditionally inhabited by the most recently joined couple from both families.
If they're merging because they share some historical origin, the new name is probably going to be a reference to that shared history, very likely a riff on an already established demonym that applies to people of that origin. If they think their distant ancestors all originally came to this land from some specific place, the new name will be a reference to that place. If they all spoke a specific language, the new name will be a reference to that linguistic tradition.
How will the newly formed nation be organized politically?
Consider the United States of America. "United States" is not a sui generis name, but a factual description of the high-level political structure of the nation: the U.S. is a collection of individual states that are still legally and politically distinct despite being close allies. When the South later seceded to form the Confederate States of America, they could have invented a completely unique name for themselves, but they didn't: they still styled themselves "States of America" because they intended to retain the large-scale political organization of the U.S. and they were obviously still on the American continent.
The same goes for the USSR: it wasn't some creative name like "Wendy," but a literal statement of the nation's political origin, organization, and philosophy.
In your case, you've got two separate countries, and each of them has its own political structure before the merger. Will the new country have the same organization, or is it changing (and, indeed, is the change the reason for the merger)?
Are they both monarchies? Will the new country also be a monarchy? Or will it have multiple co-rulers like the Roman Empire did during the Tetrarchy? In that case, the new country name might make mention of the fact.
If the new country will be a direct democracy, the new name will probably reflect that fact. Very possibly, the name will be chosen democratically, which means the new nation might be named Boaty McBoatface as a deliberate flex.
As I said at the outset, people will take the naming very seriously. That's because the name of this country is necessarily going to come up in all of the most serious contexts that are possible.
It will be printed on the letterhead of the most powerful and wealthy people who live there. It will be shouted loudly by those same people at official occasions of pomp and circumstance. It will appear in the names of countless items of national identity like flags, songs, seals, etc. It will have to be stated explicitly in every single declaration of war and peace it ever issues. People will die with this name on their lips, clutching its flag to their breast having given their life's full measure in its defense.
None of those people are going to like it if that name is some cringeworthy darling, a mashup of edgy memes, or obviously the product of lazy word association. They will want a name worthy of some dignity.