Yes, you add 1d6 from Hunter's Mark
The Hunter's Mark spell description states:
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.
Usually, a creature has one AC and one pool of hit points. The Roper is unusual because of its Grasping Tendrils feature:
The roper can have up to six tendrils at a time. Each tendril can be attacked (AC 20; 10 hit points; immunity to poison and psychic damage). Destroying a tendril deals no damage to the roper, which can extrude a replacement tendril on its next turn.
It breaks with the typical design of creatures but the Tendril feature does not specify that they should be treated as separate creatures. Contrast this with the Black Pudding's Split ability, which says (emphasis mine):
When a pudding that is Medium or larger is subjected to lightning or slashing damage, it splits into two new puddings if it has at least 10 hit points. Each new pudding has hit points equal to half the original pudding’s, rounded down.
Therefore, the bonus damage from Hunter's Mark applies to the Roper's body and tendrils alike.
You get the bonus damage
The bonus damage happens when you hit the creature. The tentacles are part of the creature notwithstanding their own hp pool. You get the bonus damage.