I think the first time I ever heard of a paladin was when visiting a friend's house, where he was playing Quest for Glory, a somewhat more RPG-like variation on Sierra's King's Quest-style adventure games. The Paladin could be unlocked in one of the earlier games, and you could carry over a save file between games to unlock it in subsequent games (the only one I actually played myself, Quest for Glory V, allowed you to simply pick the class at character creation).
Probably what really cemented my image of what a Paladin should be, though, was World of Warcraft. My very first WoW character was a Paladin, and after I re-created him on the actual server my college roommate was on (back when servers were a pretty hard limit on what other players you could interact with) he became my main (other than a brief time where my Undead Rogue was the first to get to 70 during BC).
And notably, in WoW, Paladins either fight with a big two-handed weapon or with a weapon and a shield.
5E reinforced this idea because the Fighting Styles that Paladins gained access to included Great Weapon Fighter and Protection, but not Two Weapon Fighitng. You could dual-wield, but the game was steering you away from this.
Now, I still think it's pretty effective to use this fighting style with the 2014 rules - you won't get to add that (likely +5) bonus to your off-hand attacks, which hurts, but getting Improved Divine Smite at level 11 is a pretty strong incentive to maximize the number of attacks you're making each turn.
But in 2024, I think a couple of changes are going to really make you consider dual-wielding on a Paladin. It might be the best option.
Let's talk about the factors at play:
First off, Paladins will be able to get the Two Weapon Fighting Style Feat, as all fighting styles are available to them (I think - I can't remember if Divine Warrior and Druidic Warrior made it in for Paladins and Rangers, respectively). In terms of damage gained per turn, Two Weapon Fighting is probably the best once you have your Strength (or whatever ability you're using for attacks) maxed out, as it'll give you 5 more damage compared with 4 from Dueling if you're level 5 or higher, or the rather pathetic bonus you get from Great Weapon Fighting.
(As an aside, apparently GWF is even worse now - rather than letting you reroll 1s and 2s on your damage dice, you simply count them as 3s. In other words, a d6 goes from having an average of 4 and 1/6th to 4, meaning at absolute best, it's increasing damage of a Maul or Greatsword by 1 damage per attack.)
Second, we have the Nick property from Weapon Mastery. This frees up our bonus action when we're wielding two light weapons and one of them has Nick. Given the change to Divine Smite, we'll want to keep that free (though I also think we're going to see Paladins using their spell slots for other things more often now).
Third, we have the new Dual Wielder feat. On its surface, this feat looked like it was taking a nerf - for one thing, it required that your first attack was with a light weapon, while the old one let you dual-wield longswords or other non-light weapons. However, what's notable here is that the bonus action attack granted by the new version of this feat is an entirely separate entity within the action economy from the attack granted by the light property - meaning that now that we've freed up our bonus action, we can make both that ordinary light weapon off-hand attack as part of our action and then use our bonus action to make another attack.
Lastly (maybe - honestly there are probably other factors at play) we have the newly redesigned Divine Favor, which (to the shock and horror of Rangers) no longer takes concentration.
Putting this all together, by level 5, if we're wielding, say, a Shortsword in our main hand and a Scimitar in our off-hand, and we have the dual-wielder feat and the two weapon fighting feat (which we get for free at level 2 as before) we're going to first cast Divine Favor to add 1d4 radiant damage to each hit. Now, we attack twice with our Shortsword, hitting for 1d6+1d4+4 (we'll assume we started with a 17 in Strength and used the Dual Wielder feat to bump that to 18) or 10 damage total, then attack a second time (with advantage thanks to Vex) for another 10. Then, we make our off-hand attack from Light, still part of our action, for another 1d6+1d4+4, or another 10. That's 30 damage this turn.
Next turn, we do the same 30 damage but then also make a bonus action attack for 6 (unfortunately, we don't get to apply the Two Weapon Fighting bonus to the Dual Wielder attack). So, 36 damage per round (if we always hit). Now, if we need a burst of damage, we can Divine Smite, which will sacrifice 1d6+1d4 to get 2d8 - clearly more damage, but not by a ton - the difference between 6 and 9 (though it goes up to 13.5 if we're hitting an undead or fiend target, and obviously we might take advantage of a crit).
To put this into perspective, let's look at how a similarly-built Paladin focusing on a big two-handed weapon would do. We'll keep it simple and go for a Maul, and rather than Dual Wielder, we're taking Great Weapon Master, which in most cases is actually a better feat than its previous iteration, despite the fears that it was a nerf.
Again, we'll cast Divine Favor, adding a d4 on top of our 2d6+4. Great Weapon Master will be adding our Proficiency Bonus to damage with our Heavy Weapons, so at this level we're throwing a 3 on top of all of that, giving us damage equal to 2d6+1d4+7, which comes out to 16.5 damage per hit (again, we're simplifying and ignoring crits).
Thus, on our first turn, our GWM is ahead, dealing 33 damage per round.
On turn two, though, we're just doing the same thing, meaning our dual-wielder is making up for their earlier deficit. In two rounds, GWM is doing 66 damage, but that's actually exactly what our Dual Wielder will have made it to on round two as well. By round 3, the Dual Wielder is ahead.
Now: another factor we haven't really touched on are the Vex and Topple/Graze Masteries.
The math here gets complex enough that it might not really be solvable in a generic situation - Topple could grant advantage on attacks by knocking foes prone, but we need to know their chance to resist the effect. Vex is going to not only make subsequent attacks more likely to hit, but also increase the chance of a critical hit. (Topple can also do this, but is less reliable than Vex in terms of granting advantage). Graze, of course, allows for guaranteed damage, but won't benefit from Divine Favor.
Now, I do think that Divine Favor is going to be an absolute staple for Paladins moving forward - it's only slightly less damage than Hex or Hunter's Mark, but without a need for concentration, it's going to be very reliable. I will say that in terms of raw damage, you will need a fight to last a decently long time to let it outperform a Divine Smite. For a 1st level spell slot and a bonus action, Divine Smite deals a guaranteed 2d8 damage, or 9 on average. We're going to need to hit four times on average for the spell slot we spend on Divine Favor to deal more than that (4d4 being an average of 10). That's actually easier accomplished with a dual-wielding build, where you can potentially get three of those attacks in on the first turn.
At level 11, Dual Wielding gets a significant boost from Radiant Strikes, the new name for Improved Divine Smite (a name change I'm very happy about, as Improved Divine Smite, ironically, didn't actually interact in any way with Divine Smite).
We might not actually have a +5 to Strength yet, as we're probably more likely to be taking general feats instead of ASIs (which, I guess, technically are also a general feat now) and I think it's highly possible we'll be taking some more powerful spell than Divine Favor, but let's say we're trying to conserve our higher-level spell slots.
If we haven't bumped our Strength all the way to 20 (and our DM is being mean and hasn't given us a +X weapon) the damage here simply increases by 1d8 for each attack.
Thus, our Dual Wielder, who was initially doing 30 damage on the first turn and 36 on subsequent turns, can add 3d8 to that first turn (13.5) and 4d8 to later ones (18). So, we'll be looking at 43.5 and then 54.
Our GWM is getting a bump from the feat, adding 1 damage to each attack, plus 1d8, so basically 5.5 more damage with each hit, for a total boost to their damage of 11 over a turn. Thus, each of our turns will be 44. In other words, even needing a turn to set up, dual-wielders are only .5 damage behind our GWM character (which is essentially negligible) and once things get going, they're doing over 10 damage more per turn than their GWM friends.
Now, once again, we have to keep in mind the caveat that our dual-wielding character is going to be using their bonus action each turn on the second off-hand attack. Thus, our GWM might be smiting more often or using other abilities. If our dual-wielder needs to skip a bonus action attack to go Lay on Hands an ally, they'll be dropping 1d6+1d4+1d8 damage, or 10.5. And our GWM character might find a way to add in some more damage - if they take Polearm Master as well, their main attack damage drops a little (replacing their 2d6 with 1d10, or a drop of about 1.5 damage per hit) but gaining their own bonus action attack (which will deal 2d4+1d8+4, or 13.5... no, wait, this is still an attack with a heavy weapon, so that's 1d4+4, plus 1d4 for Divine Favor, plus 4 for GWM, plus 1d8 for radiant strikes, so we get 16.5).
Actually, let's consider the GWM/PAM build's damage output, as it might beat our dual-wielder.
At level 11, we've gone with feats at levels 4 and 8, meaning we're not getting to 20 Strength until at least 12 (our Dual Wielder only relies on a single feat here, but we might imagine they've taken some other feat). We're maybe taking a Lance (assuming this works with Polearm Master, but we're not really worrying about advantage gained by topple here, so it could be a Glaive, Halberd, or Pike as well) so we've got a d10 weapon die. Thus, our first turn we're bonus action casting Divine Favor, then attacking twice and hitting for 1d10+1d4+1d8+4+4, so that's 20.5 overall damage on average, doubled to 41 on our first turn.
Second turn, we're adding in our pommel attack, which gives us 1d4+1d4+1d8+4+4, or 17.5 additional damage, for a total of 58.5.
And, just to re-do the math, let's now assume our dual-wielder just took the ASI feat at level 8, so they do have their Strength at 20. First turn, they cast Divine Favor, and make two shortsword attacks and one scimitar attack that each do 1d6+1d4+1d8+5, or 15.5 damage per hit, or 46.5 damage on turn 1.
Second turn, they get to throw in their Dual Wielder attack, which adds 1d6+1d4+1d8, or 10.5, for a total of 57 damage.
Interestingly, it looks like the tables turn by this point - our Dual-Wielder is less punished by their set-up round, but our GWM now deals more damage in the long run. In two rounds, DWs are doing 103.5 damage, and GWMs are doing 99.5. In other words, GWMs are behind by 4 damage, but their post-first-round damage is 1.5 damage more, meaning it will take them 3 more rounds to overtake our Dual-Wielders, which is a pretty long time given the length of most D&D combat encounters.
Again, other factors will come into play - GWM scales in a way that Dual Wielder doesn't, really - even if you account for a Dual-Wielder getting magic weapons with a +X bonus, our GWM needs only to get a +X Lance or whatever to get this bonus on all their hits. Meanwhile, GWM's feat damage bonus will go up by 1 at level 13, and again at level 17.
For the hell of it, let's look at 20th level paladins. At this stage, we're probably got a +6 to Strength thanks to Epic Boons, and we'll throw in +3 weapons just for fun. But for some reason (again, maybe we're conserving higher-level spell slots) we're still using Divine Favor (and hey, it's non-concentration, so why the hell not?)
Our Dual-Wielder is using a +3 Shortsword and a +3 Scimitar. Turn one, we're attacking for 1d6+1d4+1d8+9 three times, dealing 19.5 damage on each hit, or a total of 58.5 damage. Turn two, we're adding in our second off-hand attack for 1d6+1d4+1d8+3, or 13.5, so turn two damage is 72 damage.
Our GWM is using a +3 Lance, so turn one we're looking at 1d10+1d4+1d8+15, or 27.5 damage per hit, for 55 damage on our first turn. Second turn, we add our pommel strike which does 2d4+1d8+15, or 24.5, boosting our damage per round to 79.5.
Again, our dual-wielder has the edge on the first turn, and after two turns, we're looking at 130.5 for the DWs and 134.5 for our GWMs.
And, honestly, that's pretty darn close. This is before we factor in any advantage we might have from Vex or Topple (though this becomes less relevant if we're Vengeance, as we'll be getting advantage anyway with Vow of Enmity, which no longer requires any action to activate or even to move to other targets).
Either of these builds is going to be really nasty, I think. I will say that I expected dual-wielding to really overtake the old standard Great Weapon build. This makes some strong assumptions about taking both GWM and PAM, which will cut into your ability to boost things like your Charisma - the Dual Wield build is somewhat less feat-dependent (I think our GWM falls off significantly if they don't take both of these feats).
One more caveat: we've been using Divine Favor at all levels here. Surely, Paladins will consider using other spells as well, which might require more ramp-up time, and if they're based on adding damage when you hit (like Spirit Shroud) the Dual Wielder might start to inch a little forward.
I also think it's important that players consider not always thinking of things in terms of hitting a target dummy. One thing I'm excited about with the change to Divine Smite is that, now that the other Smite spells work in the same way (still taking a bonus action, but activating immediately when you land a hit - or crit) will encourage Paladins to think less about maximizing their own damage and instead doing more to control the battlefield. Blinding the target with a Blinding Smite might do less damage than a Divine Smite, but it could also make the foe less dangerous and give your party members a much easier time hitting them. Paladins have always had a lot of utility, but I think that the de-emphasis on Divine Smite might encourage players to explore what they can do beyond melting foes.
Still, there's a lot in the new systems that will make for a very strong dual-wielding paladin in a way that the 2014 rules didn't really support, even if it still worked. On an aesthetic level, I might still favor the GWM (also, you're less reliant on getting multiple powerful magic weapons,) but I can really picture a zealous fanatic of a paladin fighting with light weapons being a real vibe.