It does indeed appear to be a certain level of intelligence. In the case of the Falmer, they are corrupted and betrayed Snow Elves (see Knight Paladin Gelebor's description of what the Dwemer did to the Falmer: http://en.uesp.net/wiki/Skyrim:Knight-Paladin_Gelebor) who used to have black souls until the Dwemer betrayed them and did something that made their souls white so the Dwemer could use their souls for their machines. In the case of the giants, while they do use tools/have a rudimentary sort of civilization, we don't see anything more than that.

The matter is complicated by the fact that black soul gems only appeared in games after TES IV: Oblivion where they were introduced as part of Mannimarco's necromancy. So its hard to judge whether or not certain things in Morrowind like the Dagoths might have had black souls or kept the grand souls they had in game. There's a similar issue for the NPCs one can soul trap, since we would expect them to have black souls in later games. So unfortunately, on a meta level, TES III Morrowind isn't much help since the black soul gem wasn't developed as a game item yet.

Both Skyrim and Oblivion seem to use black soul gems to capture sentient, humanoid beings, including some types of higher daedra like dremora or golden saints. In meta terms, a black soul gem is used to capture the soul of an NPC, which thanks to the coding is men, mer, beastfolk, and dremora.

As for power, it appears to be related to level. As with the falmer, you'll see different levels of souls according to their level in game, but none of them ever have the sentience to reach the level of sentient humanoid being except for certain Dawnguard NPCs. The farmer is unquestionably a sentient humanoid (an NPC) and so has a black soul, and yet the giant, not being at that level but still very powerful among the non-NPC class, has a greater soul. Dead NPCs such as Olaf One-Eye and Dragon Priests have grand souls, where you would expect them to have black souls in life.

I suspect the real in-game lore reason is "We tried using various levels of soul gems on different creatures and people and these were the highest level soul gems that worked on each. This is just how the world works. The line seems to be that sentient humanoid beings have black souls." And now I think I've figured out Mannimarco's thesis presentation that got him reprimanded by the Psijic Order...

The out-of-game meta reason is that sentient humanoid beings are in the NPC class, and from Oblivion onward, black soul gems were introduced to allow players to soul trap those NPCs without messing with Azura's Star the way Morrowind did. From there, Skyrim appears to have tried to address the issue with the Falmer specifically as a race that used to have black souls was warped and abused by the Dwemer as fodder so that they only have white souls of varying levels now. Giants, hagravens, and undead NPCs seem to be in the same boat of being somewhat humanoid and somewhat intelligent, but not reaching the level of sentient humanoid that would get them NPC status and thus a black soul. (Now, there's a whole debate to be had over whether this is fantastic racism at work here in-game or on the developers' parts such as with Moira the Hagraven from the quest A Night To Remember, but that doesn't really shed any additional light on what soul gems work and which don't.)

Personally I'd like it has something to do with spiritual proximity to the Daedra/Aedra or another divine being.

Souls are power, much like magicka in a sense, given that a captured soul can be used to power a spell. However, access to magicka doesn't mean that a soul will be black in of itself, as shown by Falmer and Spriggans.

My hypothesis is that the distinction comes from how much power the being was imbued with from the gods or whatever you wish to call them.
In the case of Giants, their gods 'died' and they lost part of themselves, devolving into what you see now. And for the Falmer, that connection was severed by the Dwemmer in some sort of experiment.

The Mage's Guild Missive is a Crock, and proof that ZOS really doens't understand the lore and where things actually come from. The whole being able to trap sentient mortal souls in standard Soul gems is a HUGE blunder on the lore.

1. Where did Black Soul Gems come from?

the Answer is The Ideal Masters.

2. Who are the Ideal Masters?

"The Ideal Masters originated on Nirn in the Merethic Era as an early order of sorcerers who practiced necromancy and trafficked in souls. They grew to become very powerful, and eventually found their physical forms to be unacceptably weak and limiting. After transcending these forms and becoming beings of pure soul-energy, the sorcerers entered Oblivion as immortals and selected an area of chaotic creatia to create the Soul Cairn, a realm ideal for their role as soul merchants.Although immortal, they are reluctant to expend magicka as this "diminishes their eternity".

3. Prior to Mannimarco's Ascension at the end of TES II:Daggerfall(More on that later) How are Black Soul gems created and where to do they come from?

The Ideal Masters. Black Soul gems come from the Soul Cairn. They are created in Lightning Attractors by placing a Grand Soul gem inside the piece of equipment and allowing the lighting in the Soul Cairn to strike it. This lighting is the essence of the Ideal Master's corrupting the Soul Gem to allow it to bypass Arkay's Restrictions.

From Mannimarco's very own writing in TES3: Morrowind - Arkay The Enemy:

"Arkay's Blessing prevents the souls of men, beastmen, and elves from being used without consent. Arkay's Law prevents those buried with the proper rituals from being raised to serve my children's will. As you know, my children, Arkay's Blessing is flexible to those with daring, but Arkay's Law is unwavering."

Its not the Soul Trap spell that prevents or allows a Soul to be trapped, Arkay has flatout protected the sentient races on Nirn from having their Souls Trapped in Soul Gems. Mannimarco found ways around this by doing what Necromancers do, making deals with the Ideal Master's in return for power...The Ideal Masters give him corrupted Soul Gems(Black Soul Gems), He would trap Black Souls in them and trade them to the Ideal Masters for knowledge and power...its theorized that Mannimarco learned the secrets of becoming a Lich from the Ideal Masters, but thats a topic for another time.

4. How are Black Soul Gems made after Mannimarco's Ascension?

With the Dragon Break known as The Warp in the West, Mannimarco ascended to Godhood becoming the god of the Necromancer, and a new moon appeared in the sky, the Necromancer's Moon, . Since Mannimacro succeeded in one of the possibilities, he succeeded in all of them. He left a weaker avatar of himself behind that was later killed by the Champion of Cyrodiil in the 3rd era, but Mannimarco is not truly dead...He will come back again, because he is the God of Liches pretty much.

That being said, on certain dates the "Shade of Reverent" takes place(Mannimacro's influence), and people can put Grand Soul gems into altars that catch this Shade of Light from the Necromancer's Moon and turn Standard Soul Gems Black whihc allows Mannimarco's followers to bypass Arkay's Restrictions on trapping of Sentient Souls.

As i said, the spell itself has nothing to do with it. Arkay actively prevents the souls of sentient mortals being from being trapped. Hence why Manimacro had to deal with the Ideal Master's early on to get around that, and later became the God of the Necromancer to help his followers get around it.

Even Azura's Star is not able to capture the souls of Sentient Mortals. In Skyrim Azura's Star was corrupted with Necromantic Magic, something similar to how standard soul gems can be turned black, but this is a temporary condition as the star will eventually fade back to oblivion and return to its normal state.

Mannimacro and Gallerion both studied together on the Isle of Arteaum. Mannimarco got kicked out for his dark dealings. At the time, even the Psyjiic;s knew the only way to trap sentient souls was to deal with the Ideal Masters who most certinaly at some point the asking price would be too high and they would trick you and take your soul at some point.

Black Soul gems are much more common in the 4th era due to the fall of the Mage's Guild, no real restirctions on Necromancy, and much more access and many more followers who use the Shade of the Reverent.

http://en.uesp.net/wiki/Morrowind:Arkay_the_Enemy

http://elderscrolls.wikia.com/wiki/Lightning_Attractor#Lightning_Attractor

http://en.uesp.net/wiki/Lore:Ideal_Masters

And here i thought "black soul gem" means the soul gem is black, not the soul