The tentacle rod's description includes the following (emphasis added):
Made by the drow, this rod is a magic weapon that ends in three rubbery tentacles. While holding the rod, you can use an action to direct each tentacle to attack a creature you can see within 15 feet of you. Each tentacle makes a melee attack roll with a +9 bonus. On a hit, the tentacle deals 1d6 bludgeoning damage.
Answering this recent question, I concluded that the three-attack feature of the tentacle rod is incompatible with extra attacks from its user. In this older question, it was concluded that it is also incompatible with opportunity attacks.
Usually, magic weapons have the respective item category (e.g. the scimitar of speed, which says "weapon (scimitar)") that defines damage based on those weapon types detailed in the PHB (p. 149). The description clearly states that the rod is a weapon (vide infra), its category, however, is rod. Rods, contrary to staffs, cannot usually be used as quarterstaffs (see this question about rod of the pact keeper).
It seems, therefore, that the tentacle rod is a weapon by RAW but has no weapon type or weapon damage defined.
The rules on improvised weapons (PHB pp. 147-148) state:
An improvised weapon includes any weapon you can wield in one or two hands, such as broken glass, a table leg, a wagon wheel, or a dead goblin. Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club.
It further states:
An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object).
Now the last part should not apply because the rod clearly is a weapon. I can simply decide what weapon the rod is like (probably a quarterstaff, club, or a whip).
TL;DR
I could not find conclusive information on what the damage would be when making a normal weapon attack with a tentacle rod.
Is there anything more clear on the subject of the amount of damage, either additional rules on improvised weapons, or specific information on the tentacle rod, that would give guidance on the matter?
Best Answer
I can find nothing in the description of magical rods in general, or the tentacle rod in particular, that defines its weapon properties.
... in 5e. It may be worth noting that 3.5e’s description of the physical properties of magic rods said
This obviously does not apply directly to 5e, and even if it does it still leaves a question of whether the tentacle rod should behave like a club or a light mace, but it does give precedent to the idea that if you wield a magical rod, even if you are not or even cannot activate its magic powers, you can always hit ’em with it. That, to me, speaks to “weapon” being used correctly in the description here, not as very-poorly-chosen descriptor.
I would not, personally, even rule it as an improvised weapon—it says that rods have “sturdy construction” that allows them to “function as light maces or clubs.” I would rule it as a fancy club or light mace. It may not matter, but if for any reason you are better with clubs or light maces than you are with some random improvised weapon, I’d let you have it.