What's the origin of this approach of trying to obtain meaning from the individual letters of Hebrew words? In a recent question about the names of God, two answers from different people wrote that the letters of the Tetragrammaton by themselves give the meaning of:

The Hebrew letters mean “Behold the hand, Behold the nail”. YHWH is our salvation.
Yud. (A hand)
Heh. (Behold)
Waw. (A nail)
Heh. (Behold)

(From one answer, here's the other.)
Another now deleted answer says that the Hebrew word for 'ten' has these symbolic letter meanings:

The first character is the `ayin, ע, and symbolically means to see, eye, discern, or divine providence.
The second character is the Shin, ש, and symbolically means the tree of life, burning bush, God’s spirit, etc.
The third character is the Resh, ר, and symbolically means a humble or penitent man like he is bending over in prayer.

Now at first glance this seems to me to be pseudo-linguistic nonsense. Spoken language precedes written language, and the meanings of words aren't derived from their sounds or letters. These &quot;meanings&quot; also seem pretty speculative. They don't have the historical evidence that gematria has, and most aren't based on the historic origins of these letters (except for Yod and Ayin):

Waw/vav came from a glyph for 'mace'
Resh comes from a glyph for 'head' (and the word ראש still has that meaning!)
Shin apparently came from a glyph for 'tooth'

But even if the rest of these letters did have the symbolic meanings claimed about them, so what? It's just not how language works! Words have their own meanings, often distinct from the parts that make them up. We all know that 'awful' doesn't mean 'full of awe' in modern English, even if that was its historic origin.
So where has this approach to Hebrew come from?

Do any branches of Judaism support this approach?
What is its historic origin?
And lastly, is there a name for this theory?
Psalm 119 may have been an influence, the presentation of each Hebrew letter heading the twenty two sections of the Psalm. I have always taken this as a demonstration of reverence for the word of God, the singling out of each letter as being important in the whole structure of scripture.

But perhaps meaning has been seen in each section of the Psalm that has been attached to the individual letters. It would be interesting to examine the psalm in detail to see if the meanings imposed, as in the OP, are taken from the Psalm.

צ (Tzaddi) the eighteenth letter of the Hebrew alphabet, heads the eighteenth section of Psalm 119 and is also the letter which is at the beginning of the Hebrew word for righteous, saddiq צַדִּ֣יק which is the first word of that section of the psalm.

So perhaps - in scripture - the letter tzaddi stands for righteousness.

There is much more to Hebrew scripture than the English translation displays. Lamentations, for example, is highly structured poetically. The first four of five poems are written as acrostics – chapters 1, 2, and 4 each have 22 verses, corresponding to the 22 letters of the Hebrew alphabet, the first lines beginning with the first letter of the alphabet, the second with the second letter, and so on see Wikipedia - Lamentations.

Even in his grief at the state of Jerusalem, Jeremiah poems his lament in a highly structured, artistic and intelligent expression of reverence towards God.