At level 3, a Path of the Giant barbarian can become Large while raging. How does the damage change?
It's just that creatures with the size of a large Troll or Ogre type inflict 2dX. I mean, the bigger the creatures, the more damage they cause.
How much damage will the character do when it gets Large or Huge? Nothing is written about it. I have assumptions that it will do 2dX when Large and 3dX when Huge.
The Path of the Giant feature says:
Giant Stature. Your reach increases by 5 feet, and if you are smaller than Large, you become Large, along with anything you are wearing. If there isn’t enough room for you to increase your size, your size doesn’t change.
As you observe, these rules do not mention a damage increase of any kind. And as there are also no general rules for increased damage for large creatures in the game, this means as written the feature has no effect on damage. They also do not increase in Strength, even though one should think their much larger muscles can lift more.
(This is different in some other games, for example in Call of Cthulhu melee damage is directly tied to character size.)
The closest you get in player facing rules in 5e is the Enlarge spell, which says it adds a meager +1d4 damage, but that is a specific effect of the spell, not a general rule.
In the DMG, for statting Monsters, there is guidance to double the dice for large, triple them for giant creatures. And one can see that in many monsters, for example the large Ogre deals 2d8 + Strength bonus with its greatclub, instead of 1d8. But again, this is a guidance on how to stat monsters, not a general rule that players can use.
If you feel that no damage increase for becoming large is hard to swallow or justify, then you as the DM are free to overrule the rules, using one of the above approaches, or one of your own invention. Just keep in mind that the features are balanced as they are.
Nothing is written about it
Then nothing changes about your damage dice. Giant’s Havoc states:
While raging, you gain the following benefits: […]
What comes next is everything that changes about the barbarian’s base Rage feature. Notably, something to the effect of “increase damage dice to NdX” is absent. Thus, this feature does not change your damage dice. It does exactly and only what it says it does. You don’t have to look elsewhere to know what your class features do.
For a more detailed explanation, see my answer about the monster homebrew guidance in the DMG. The short version is that there is no general rule for players that increasing size increases damage.
