The Half-Dragon Template in D&D 5e (MM, p. 180) states that, "The half-dragon has the breath weapon of its dragon half." However, the sample statblock given for a half-red dragon veteran has a different save DC (15) than the red dragon wyrmling (13) it is based off of. I have attempted to find reasoning behind this, though I haven't been able to find it through set mechanical standards.
Why is the save DC for the half-red dragon veteran example's breath weapon listed as 15 instead of 13?
I suspect it's a mistake. But if it is, the correct DC should be not 15, not 13, but 12. Yes! Twelve!
We know from the PHB that the DC for the Dragonborn's breath weapon is 8 plus the user's proficiency bonus, plus the user's Constitution modifier, making Constitution the breath weapon's "spellcasting ability." (Though it's plenty obvious, I should clarify that it's not a spell.)
The same calculation is used for the various dragons' breath weapons. You can infer a dragon's proficiency bonus by subtracting the damage bonus from the to-hit bonus of, say, its Bite attack. The Red Dragon Wyrmling's proficiency bonus is apparently +2; the DC for its breath weapon is 13; 13-8-2 = 3, and +3 is the Wyrmling's Constitution modifier.
This actually applies to the spell-like abilities (I use the phrase in a non-technical sense) of every creature found in the Monster Manual. If you do the math, the save DCs for the harpy's song, the cockatrice's bite, and the rust monster's antennae are all based on their respective Constitution modifiers.
Except for the breath weapon of the example Half-Red Dragon Veteran. Its proficiency bonus, based on its various weapon attacks, is clearly +2. Its CON mod is +2. The save DC for its breath weapon should be 8+2+2 = 12.
But the book says it's 15, and that doesn't really make sense. It may have been assigned arbitrarily to best fit the intended Challenge Rating of 5, but this would be odd, considering how consistently the Constitution-based calculation is applied elsewhere. It may have been calculated based on a much higher Constitution score or proficiency bonus, and then got overlooked in a revision when the Half-Dragon got powered down. It may have been a simple typo.
For all I know, the example Half-Dragon was extensively playtested with the printed numbers, and the number 15 was arrived at after months of careful balancing. But if we play by the (inferred, but consistent) rules of the Monster Manual, the breath weapon's save DC should be 12.
As you pointed out, copying the save DC from the Red Dragon Wyrmling would result in a DC of 13, and as the other answer pointed out using the DC calculation used for Dragons would result in either a 13 or 12, depending on if you used the Proficiency bonus for the Half-Dragon Veteran or the original Veteran.
Both of which would not result in the 15, so neither can be how they calculated it, and because they haven't changed it in almost 10 years, it most likely isn't a typo either. So how did they end up with a DC of 15? As Ryan stated: "It may have been assigned arbitrarily to best fit the intended Challenge Rating of 5". Which is the only possible explanation left, well kind of. The DC of 15 isn't just an arbitrary number, it is infact the default save DC the Dungeons Master's Guide suggests for a CR 5 monster.
On page 274 of the DMG you can find a table for quick monster stats, listing default suggestions for a monster's save DC, among other things. And as stated above, it just so happens that a CR 5 monster gets a DC of 15. So it would seem, that the game designers intented for us to pick the Save DC for the breath weapon from this table whenever we use the half-dragon template on a creature. The fact that they omited any kind of equation to calculate the Save DC, as they did for the Dragonborn race for example, also supports this theory.
The Young Red Shadow Dragon, another creature in the MM that was created through the use of a template, further supports this theory, as it also uses the save DC appropriate for its CR from that table. It has a CR of 13 and a save DC of 18, which is different from the regular Young Red Dragon's save DC of 17. One could argue that this the result of its proficiency bonus increasing with its CR, however we can cleary see that its PB has not increased, as it still uses the same attack bonus and save bonuses that the regular Young Red Dragon does. If its PB had increased, all of those would have to be 1 point higher.
Back to the half-dragon: I can only assume, they didn't want creatures with high Constitution values to get fairly high Save DCs, as that might increase their CRs even further and monsters with low Constitution values, to have a Breath Weapon with a CR so low, that it's basically useless. But yeah, this really is a confusing stat block. Why use the default value here, but not for the original dragon or even the half-dragon's weapon attacks? And also why use the DC values for the new CR, but then the proficiency bonus for the old CR? Let's all hope, they make half-dragons a whole lot less complicated in the revised Monster Manual coming in 2025.
