Sentient Phlebotinum

I am creation and destruction. I am the power to transform and to destroy. I am every drop of Energized Protodermis that exists, and every drop is me. I am as far beyond you, creature of armor and tissue, as you are beyond an insect.

—Energized Protodermis, Bionicle

That Applied Phlebotinum you've known about and have been using since ever? It's alive ... question is, what are its intentions?

If it's also biological it may also be a Living Battery. May or may not be an Empathic Weapon or Living Weapon. See also Powered by a Forsaken Child and Artifact of Doom.

Examples of Sentient Phlebotinum include:

Anime and Manga

Innocence from D.Gray-man; its implied/stated how Innocence "chooses" people or something like that.
The Hougyouku from Bleach.
The Shikon No Tama from Inuyasha
The Paksis in Vandread.
Getter Rays from (what else) Getter Robo are either this or a Cosmic Horror.
The Evangelions.

Film

Flubber
The Crystal Skull from the Indiana Jones and the Kingdom of the Crystal Skull.
The angels in the various angel sports movies: e.g. Angels in the Outfield.
The antagonist of Sci Fi movie Fire From Below. It is a radioactive substance, useful for both a power source and armor plating when depleted. The problem is it seeks certain substances to burn with virtual sentience.

Literature

Dust in His Dark Materials.
Simon R. Green's Secret Histories series concerns this, first with the family's enchanted armor is powered by a evil fugitive extradimensional being that eats newborns. The replacement is possibly a good guy, but being a Starfish Alien it's not easy to tell its complete motives.
In The Magic Thief series, by Sarah Prineas, Conn discovers that the magic of Wellmet is actually a living being, and that spells are its language.

Live Action TV

Doctor Who: you see that blue police box that can travel through time? Well, it's not just the Doctor's space-ship. 'It' is a 'she', and often chooses where the Doctor lands.

Toys

Bionicle: Energized Protodermis and Antidermis.

Video Games

Metroid Prime: Phazon. It also happens to be evil.
Metatron in Zone of the Enders, the resident Phlebotinum, is at least Psycho Serum and possibly sentient. Without an AI intermediary to "filter" a pilot and metatron the pilot tends to go insane and (in the above case) construct a virtual ghost of a dead lover to embody their genocidal tendencies.
The lums in Rayman. They're small orbs of energy which usually float towards you. The Backstory in Rayman 2 reveals that they combined their collected conciousness to create Polokus, who would then create the world.
In Freedom Force Versus The Third Reich, Energy X actually turns out to be Sentient Phlebotinum. And not exactly a nice entity, either.
Gears of War 3: Immulsion, which is a parasitic lifeform.
The magic cloth in Journey appears to have sentience and, in fact, different species within itself, from small scraps that float through the air together like fish, to a more familiar tassled Flying Carpet style, to a massive dragon-looking creature. They emit tones just like the traveler, and aid you on your travels.

Western Animation

Generator Rex: Alpha, the original control nanite. He was made sentient to be able to properly control the nanites to do tasks, but that didn't end too well. Any body he makes for himself is basically a swarm of nanites as well... Until he burns through them and has to get more nanites to build another new body, at any rate.
- The Meta-Nanite has some sort of intelligence since Rex could talk to it and order it to cure every Evo in the world. Well, almost every one.