Option 2 is the most balanced out of the three proposed changes in the question
The reason I originally considered this many months back is because of a LMoP game I ran, which had a Dragonborn Paladin player. We started with no adjustments to RAW, so this is an account of how RAW seemed to work at Tier 1 levels.
During the first dungeon, at level 1, that breath weapon took out 2 or 3 goblins (they all failed their saves, and he rolled good damage). The breath weapon as-written looked great at that level! However, as we progressed towards level 5, the unimpressive damage scaled poorly with the rest of the encounters, and it become objectively better in all situation to just use standard paladin tactics (including a simple "I attack with my sword") than to "waste" his action with the breath weapon.
(I will point out that this is a well optimised paladin, with 20 STR and 18 CHA at level 1 (!) and the Dueling fighting style; that plus a +1 weapon means damage output is 8 minimum per hit in melee. His CON is average at about 14 or 16 or so–I forget exactly–which I will admit doesn't help as much with the DC the enemy need to make saves against to halve the breath weapon damage).
The anecdote above about the goblins is precisely why I think my "option 1" is too strong; 2d6 is plenty for a level 1 character, but by level 3, that's when we started to see it become less useful (low damage rolls and high saving throws didn't help, but still). This is why I suggested in the question (as my "option 2") increasing the damage at 3rd level by 1d6 and allowing that increase to propagate throughout the RAW damage increases.
More recently, I trialled my "option 2" rule with a different, higher level Dragonborn Paladin in a different game I'm running, to see how it compared with the Dragonborn Paladin in the game I described above. However, although the breath weapon didn't feel as weak now, it was still better most of the time to just make melee weapon attacks instead, which was more apparent with the addition of Extra Attack at 5th level. That's when I identified the real problem with the breath weapon...
The Breath Weapon directly competes with Extra Attack
Because the breath weapon takes up a whole action, but isn't the Attack action, it means that's all you can do. Given how heavily weighted towards Paladins the Dragonborn race is (due to +2 STR, +1 CHA), the fact that the breath weapon will quickly become useless for a martial class is the main problem, even more so than the relatively small amount of damage it can deal. During Tier 1, this isn't really a problem because they don't have Extra Attack yet, but as soon as they get Extra Attack, its always the superior option to attack with your standard weapons.
So, in addition to going with my "option 2" damage increase, I also added the following houserule (on the fly, after a disappointing use of the breath weapon): "Using the breath weapon now counts as an attack (same as how shove or grapple is a special kind of attack), meaning that it can be used alongside Extra Attack". Now the breath weapon feels more useful; a Dragonborn paladin can attack with their weapon (once) and use their breath weapon during the same turn.
The fact that the breath weapon can still only be used once per rest means that this didn't become a too-frequently used combo, but at the same time, it helped the player to still feel like they were a Dragonborn who can use the breath weapon, without essentially "wasting" their turn just to use it. It's now possible to do a small amount of damage to a spread of enemies whilst actively engaging with one of them like a Paladin should, allowing others to use toll the dead and such on the other now-injured enemies.
That said, it still felt like a niche combo and a lot of the time the Dragonborn Paladin still just used standard weapon attacks, so it simply provides another option for the Paladin to use in specific situations rather than becoming a strictly superior choice, and the fact that the breath weapon still wasn't used that frequently also convinced me that this houserule is balanced, even with the slightly increased damage as per my "option 2".
The slightly increased damage and synergy with Extra Attack helps to bring the Dragonborn race up to be on par with other races
My final conclusion, then, is that, although the slight damage increase is nice, and classes that don't get Extra Attack (likely Dragonborn Sorcerers and Warlocks if they want to make use of their racial ability score increases) will get some use out of that, Paladins and other martial classes (or specifically Pact of the Blade Warlocks with the Thirsting Blade invocation) need this ability to complement Extra Attack before it will really be that useful to them.
As I said above, I didn't notice these changes making the Dragonborn Paladin in my latter game any more powerful than how powerful I would expect any Paladin would be at that level, and it simply gave that particular Paladin an extra option to use in niche situations to remind herself she was playing as a Dragonborn. This is, to my mind, what racial abilities are all about; they're are extra tools to use in niche scenarios to make players say "oh wait, I'm a [insert race here], so I can just do this!".
Although some of this assessment would equally apply to my "option 3" at these levels, I think my option 3 scales too quickly. 6d6 at level 16 should be fine, like a "mini-fireball", whereas 8d6 at level 16 sees any Dragonborn character actually being able to effectively cast fireball (albeit with a slightly different range/shape, but in terms of damage, it's an 8d6 AOE). That seems too strong for a racial ability; no other race can produce that must damage via racial abilities alone. After all, the Dragonborn aren't that underpowered, they don't need a massive damage increase, just enough for it to remain useful at higher levels...