No

The rules

The main reason for this is that the Tentacle Rod makes melee attacks, while Divine Smite requires melee weapon attacks.

Melee attacks (PHB p. 195)

Used in hand-to-hand combat, a melee attack allows you to attack a foe within your reach. A melee attack typically uses a handheld weapon such as a sword, a warhammer, or an axe. A typical monster makes a melee attack when it strikes with its claws, horns, teeth, tentacles, or other body part. A few spells also involve making a melee attack.

So melee attacks are considered melee weapon attacks only when made with a handheld weapon, otherwise they are plain melee attacks (coincidentally, tentacles are actually given as an example of this case).

The difference between the different types of attacks is very well explained by Dale M in his answer to a different question, but to reiterate:

There are 2 axes of attacks: every attack is "weapon", "spell" or "one of the specifically listed things that are neither (e.g. unarmed, grapple)" and "melee" or "ranged".

Additional musings

Tentacle rod

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 [...]

Given that the attack is made by the tentacle itself, while the PC uses his/her action to use (command) the rod (which isn't a function of the Use an object action - DMG p. 141) and not take the Attack action, this could also be argued that disqualifies the attacks for the Divine Smite requirement, as you have to be the one making the attacks:

Divine Smite

Starting at 2nd level, when you hit a creature with a melee weapon attack[...]