So I have a story about a world where honor is the most important thing and where if someone insults you... duel to the death. Most often this is peacefully resolved but when duels happen I want a kind of Wild West theme with revolvers and the like. So with 21st century gun tech why would people still duel with revolvers?
Sticking to the good old tradition: we are not punks who fight with laser, plasma annihilators and all that garbage used by these honor-less two legged beings who dare calling themselves men.
Our ancestors, those were real men, defended their land and honor with guns and lead, and so we will do.
The answer by The Square-Cube Law sounds good, but let me add tradition to the mix.
Honor can only be satisfied by a duel which meets the traditional forms. Traditional in your world are swords, and muzzle-loading pistols, and specific models of six-shooters. There are elaborate rituals of challenge and acceptance, selection of site and selection of weapons.
What is the difference between killing in a duel (lawful or at least not prosecuted in your world) and plain murder? Tradition! Anybody who shows up with a semi-automatic would demonstrate contempt for traditions and honor. It would instantly turn the attempted duel into a murder.
Your setting has to decide if the distinction is written into law (any law that is written down can be searched for loopholes) or unwritten (any cop who is not a gentleman might arrest a duelist).
Laws and regulations
If you try to duel with AK-47's or M16's you are prone to more collateral damage. And don't get me started on UZI's.
And that's just late 20th century tech. 21st century weaponry is bound to increase someone's capacity to take down a lot of people more efficiently.
Revolvers, on the other hand, are 19th century tech. The first patent for a revolver is from 1818. They don't have too great a fire rate when compared to a fully automatic 21st century weapon and it's extremely hard to modify them to fire faster. They also take long to reload even if you cheat with those reloading accessories. So the odds of someone going spree after a duel are smaller.
Deliberate range limitation
If a duel is all about "honour" then expanding the scope by killing or injuring innocent bystanders or brassing up their property is unacceptable (nod to The Square-Cube Law's post). Cutting loose with pistol rounds, even nineteenth century types, endangers people for hundreds of metres around, while rifle rounds can go 3+ kilometres (7.62 NATO).
The solution is to use an ammunition / firearm combination that will not endanger bystanders or property. Use a round that consists of tiny, 1 mm diameter pellets of shot in a gel suspension that will disintegrate after it has travelled about 10 metres (or whatever the standard duelling distance is). Fire the round at low velocity but make it a very large calibre round, say .60 or thereabouts. At short range - ie before it reaches disintegration distance - it will blow big holes in anyone it hits. However, once it disintegrates, the square-cube law (the principle, not the user on this site!) means that the air resistance on each pellet will be very high relative to its mass, so they will each decelerate very rapidly in the air to non-harmful speeds. Spectators should probably still wear goggles to avoid eye injuries. Note that this is why shotguns firing very small shot are devastating at very short range but quite safe at surprisingly short distances. (I vaguely recall reading about a weapon / ammunition combination like this for use by sky marshals dealing with hijackers to avoid puncturing the hull or windows and depressurising airliners, but cannot find a reference.)
Trying to make a large, very low velocity round cycle a semi-automatic reliably would be very challenging. This is not an issue with a revolver, though, as the cycling is achieved through pressure on the trigger (double action) or manually cocking the hammer (single action). The revolver would be very bulky due to the diameter of the rounds, so possibly a 4-shot model would be standard for everyday carriage rather than a 6-shot model.
The nature of these rounds would mean that modern body armour would be quite effective against them. Duels would require that participants were unarmoured.
If you don't necessarily have to stick with the "to the death"-rule, which would have completely depopulated our real-world-nobility very quickly, you can take the the real-world-explanation:
The point of duels was not trying to kill each other, but rather to impose a certain risk and danger on the act of insulting someone elses honour. Dueling pistols therefore have always been fairly inaccurate and archaic technologywise (for example refraining from rifling the barrel, decreasing accuracy), significantly reducing the chance of a fatal outcome.
Funnily enough even so dueling became such a problem that most nations and militaries outlawed it during the course of the 19th century because the cost of losing officers and nobleman to pointless feuds was still deemed to high. Duels to the death would have been completely unsustainable.
Fighting with honour and choosing to duel with revolver type weapons are two different things.
I would think a using a revolver type weapon when other more modern or useful are available, would be a personal choice.
The aspect of an honourable battle for cowboys is the same as samurais, this can be seen in the their movies as they share the same morals but what is more important is their reason for this style.
Both cultures the Americans and japanese lived in a lawless fashion at the times. Seeing your loved ones or local villages raped or brutalised by thugs with no morals or constant underhand tactics was the norm and any kind hearted civilian was used to this immoral and brutal life.
So those that trained hard with experts in either shooting or sword fighting learned a way that above a sneaky or thuggish way of fighting and their skills were so high that they knew they were above unhand tactics. At such a high level they realize they could easily defeat these local thugs so they only respect those others that are as high level as they are.
So it becomes a game of respect and sportsmanship, you only draw your gun or raise your sword when or your opponent is worthy and when high level fighters meet, it is all about the fight and not disrespecting them or others.
So if it is just about using revolvers I think that is a personal choice with a nod to nostalgia but if there needs to be a reason for this type of one on one life and death respect battle, then possibly they could have a back story where they lived in a lawless ruthless time and those that trained hard and became experts, even on opposite sides still respected each other in battle due to their high level.
Calibre availability, assuming modern weapons means modern armour too then you want to use the biggest bullets possible in a duel to the death. Large calibre pistols tend to be revolvers and in fact small capacity revolvers at that, mostly 4 or 5 shots for 50 cal pistol rounds and if you wanted to use 50 BMG ammo that's a single shot pistol.
Practically trial by combat is going to risk injury or death, so some form of judicial sanction is required - and a standard form of duelling keeps things 'fair' to an extent. Your average cowhand isn't going to have an advantage over a well trained, practiced professional duelist - but between equals it would be fair.
Since both sides are using the same weapons - its also possible to ensure one side doesn't cheat (by having weapon-selection randomised, or letting the challenged pick his weapon first). Rather than having the rule for a specific weapon type, they might be around calibre, and round count or that a weapon is single action.
Grayson's reasoning for katanas for duelling would be useful to copy here - that the culture of the place idolised westerns, or modelled themselves on it, so when trial by combat became legal, that was their first choice.
After the apocalypse caused by the third and massively fatal wave of the mutated Covid19 virus, civilisation has collapsed. The only survivors are survivalists. It is well know that they stock up on old-fashioned weapons.
Blacksmiths among them continue to forge these more primitive weapons and making gunpowder is still understood. All the fancy stuff, like lasers, stop working when they break down and there are no spare parts and no mains electricity to keep them charged.
Survivalists have a frontiers-person mentality and so they will naturally settle serious arguments with gunfights.
A revolver has a known small number of bullets and a known firing rate and isn't a real combat weapon. If you only have a revolver, turning the duel into a battle puts you at disadvantage.
Laws ban the use of more effective weapons in duels, so you can't use the excuse of "I am just ready to duel" to carry a more effective weapon.
Legal dueling weapons are also peace bonded; when you use them, they broadcast and record their use, so using them for common murder is impractical. The peace bond is just a broadcast, so you can use it in self defence.
Carrying a revolver means you are ready to defend your honour. Carrying a rifle or semiautomatic weapon means you are deplorable low class scum.
Modern weapons require no skill to use.
With a gyro-stabilized, recoil-compensated, AI smart targeting enabled mid-21st century smart gun, any moron can score a kill on 5 km range. Using modern firearms would make duels too easy. But it takes real skill to hit someone with a classic 19th century revolver. It does of course take even more skill with even older weapons. But flintlock pistols are so inaccurate that they require more luck than skill. And melee weapons aren't deadly enough with modern medicine, so there isn't enough risk of death involved to make the duel honorable.
In the eyes of your society, revolvers are the ideal compromise between deadliness and skill requirement.