Given the answers to these questions:
Advantage on damage rolls – is it still balanced?
'Advantage' or 'Disadvantage' on damage rolls
What should a weapon cost that grants advantage on damage rolls, but no other enhancements?
How much would a weapon cost that grants advantage on damage rolls
dnd-5ehomebrewpricing
Related Solutions
In almost all situations, improving your chance to hit is better than improving your damage roll. And thus, Attack Advantage will almost always be preferable to Damage Advantage.
I went ahead and wrote an AnyDice program to compare the two, and if you'd like to go fiddle with it, you can find it here. In the program, I compute average damage per attack for both Attack Advantage, and Damage Advantage.
I ran multiple tests with different weapons, AC bonuses, and Attack Bonuses and came to the following conclusion.
If the target's AC is more than 3 points higher than your Attack Bonus (which is usually the case), then Advantage on Attack Rolls yields a higher average damage than Advantage on Damage Rolls does. This holds up for all weapon dice-sets that exist in the PHB.
So, to give one example: that of a character with 20 STR and a Longsword...
Opposed by an AC of 15, if his Attack Bonus is +11, his Average Damage Per Attack will be...
- Attack Advantage: 9.72
- Damage Advantage: 9.45
If you increase his Attack Bonus to +12, then...
- Attack Advantage: 9.84
- Damage Advantage: 9.99
This pattern holds true as you increase AC...the larger the gap between AC and the Attack Bonus (and, in general, there will be a significant gap between the two) the less useful Damage Advantage will be.
This also follows logically. Advantage on a Damage Roll increases your chances of doing a little more damage. Advantage on an Attack Roll increases your chances of doing any damage at all. So Damage Advantage can mean the difference between doing 6 damage and 8 damage. Attack Advantage can mean the difference between 0 damage and 8 damage.
That being said, I discovered another situation in which Damage Advantage holds up better. If you do not have the two-weapon fighting feature, and so your off-hand damage does not gain the +damage from the attack stat, then the AC/Attack Bonus margin increases to 5, instead of 3. i.e. +10 to hit vs AC 15 with an off-hand weapon (no bonus), Damage Advantage is better. But, even here...it won't be often that you have such a high Attack Bonus against something with an AC that is low enough you only need to roll a 4 to hit it. So, practically speaking...this doesn't matter much.
So, TL;DR: Damage Advantage is almost always inferior to Attack Advantage.
Balancing around a once per day power is poor design
Having an infrequently used power as a balancing mechanism leads to encounters where it is used being easy for the dragonborn, and encounters where it isn't used being hard for the dragonborn. Its too all or nothing.
I don't think any amount of maths will prove that point, but if you want the breath weapon to be iconic, useful and a balancing mechanism it has to be more frequently available.
How I see it
I would like the breath weapon to be iconic, so would treat it like a cantrip. Keep the damage low (Something like D6 [Because AoE], increased every x levels like any other cantrip) but make it at will (Effectively getting a slightly nerfed burning hands cantrip).
Best Answer
101-500 gp.
Ilmari Karonen analyzed the average damage increase for such weapons here. Here are the results:
A +2 magic weapon guarantees a flat +2 increase to damage, so all of these weapons will yield an inferior damage improvement to a +2 magic weapon, without providing the bonus to hit provided by the magic weapon. +2 magic weapons are Rare magic items, and all of these would be strictly worse, so these weapons should value somewhere in the range of Uncommon magic items, which is 101-500 gp (see the "Magic Item Rarity" table in the DMG). From here, it makes sense to scale the value somewhere within this range according to the damage improvement given, possibly taking into account that the weapons are not actually magical, and so would not overcome resistance to nonmagical slashing, piercing, or bludgeoning.
Note, if a feature adds additional damage dice to an attack's damage, such as the Rogue's Sneak Attack, it should not be rolled at advantage, as the improvement can get out of hand.