Ranger casts Hunter's Mark on a werewolf and shoots it with a mundane bow. Does the werewolf take the hunter's mark damage and ignore the piercing damage from the bow, or does it take no damage at all? Hunter's Mark doesn't specify damage type like Hex, for example.
Unfortunately, by RAW, the Werewolf is immune to the damage boost from Hunter's Mark.
Like you've mentioned, Hunter's Mark does not specify a damage type, and thus the boost is of the same damage type as the weapon attack. It also does not specifically grant magical properties to the weapon attack, and thus does not allow for piercing the Werewolf's immunities.
I like to imagine the Hunter's Mark to be granting the caster the ability to place more precise strikes against the marked creature. If the creature is immune to your attack, however, it does not matter how precise you are.
To compare to a similar effect, consider the Magic Weapon spell. It makes the distinction of declaring the weapon as a "magic weapon" (in addition to the bonus to attack and damage), thus allowing it to pierce the Werewolf's immunity. If it didn't make that distinction, I'd say that the boosts (+1 to atk & dmg of the weapon) would still not get through because it'd be granting the weapon mundane "extra damage". Not all damage that is "brought about" by spell effects are magical for purposes of resistances/immunities.
Now, consider the Thorn Whip spell, which deals piercing damage without specifically mentioning that it is magical in nature. Because the damage is dealt directly by the spell (and not "extra" or "additional" damage that is consequent to a successful weapon attack), it then automatically becomes magical in nature, and would thus punch thru the Werewolf's immunity.
The Hunter's Mark damage is from a spell (i.e. magical) therefore it bypasses immunity.
Hunter's Mark states:
... Until the spell ends, you deal an extra 1d6 damage to the target whenever you hit it with a weapon attack...
The damage type is unspecified so we can safely assume it is of the same damage type as the weapon that was used. If the weapon used was a non-magical weapon, the weapon's damage would be reduced to 0 because of Immunity but the spell would do full damage, as Hunter's Mark only depends on the attack hitting the creature, as opposed to damaging the creature.
The thought process is so:
- Did you hit the creature? Yes.
- Did the weapon's hit damage the creature? No.
- Did Hunter's Mark trigger? Yes.
- Isn't Hunter's Mark damage magical because it's a spell? Why, yes. Yes, it is
- Then the Werewolf takes 1d6 damage.
If this is difficult to believe, remember that magical weapons (and silvered weapons) pierce through this immunity. So even if the damage type is the same, the difference-maker is the magical effect of the weapon. In this case, the weapon damage isn't magical but the damage from the spell is 100% magical. (thanks to @Josh Clark for finding this tweet)
Take, for example, another spell:
Your prayer empowers you with divine radiance. Until the spell ends, your weapon attacks deal an extra 1d4 radiant damage on a hit.
If you cast Divine Favor, and you hit something which has immune to radiant damage, the damage would be calculated as:
5 (1d8 mundane slashing, longsword) + 3 (mundane slashing, str mod) + 0 (1d4 radiant, from spell)
If you hit a creature with immunity to BPS damage, it is:
0 (1d8 mundane slashing, longsword) + 0 (mundane slashing, str mod) + 3 (1d4 radiant, from spell)
Notice how the damage is calculated independently of each other for the purpose of overcoming immunity.
With that said, if you hit a creature with immunity to BPS but with Hunter's Mark on it:
0 (1d8 mundane piercing, longbow) + 0 (mundane piercing, dex mod) + 4 (1d6 magical piercing, from spell)
Only if the weapon normally bypasses said immunity.
PHB. pg. 251
By RAW, the spell states (emphasis mine):
You choose a creature you can see within range and mystically mark it as your quarry. Until the spell ends, you deal an extra 1d6 damage to the target whenever you hit it with a weapon attack, and you have advantage on any Wisdom (Perception) or Wisdom (Survival) check you make to find it.
The bolded section implies, very heavily, that the extra damage is from your weapon since it is directly contingent on the weapon. This is further reinforced by the fact that unlike other damage spells, when cast at a higher level the damage remains unchanged and instead the duration is affected.
Consider this to be the same effect as a sneak attack from a rogue, in that the damage type is dependent on what is hitting the target and is not calculated separately.