What's the origin of this approach of trying to obtain meaning from the individual letters of Hebrew words?

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.

In 2010 I found this interpretation of the meaning of the Tetragrammaton as given by a Messianic Jew:

YHVH : יהוה

Yod - י hand

Hey - ה Open Window, to reveal

Vav - ו Nail / Tent peg

Hey - ה Open Window, to reveal

Meaning: The hand through which everything is revealed, the nail through the hand is the covenant sign.

However, when I went to Mi Yodea to make enquiries, I was informed that the letter Vav doesn’t mean nail or tent peg, which is actually associated with weapons of war and violence (יתד). Vav, which literally means Hook, has a connotation of joining together.

I was also informed that in Judaism everyone is free to offer their own interpretation, and there is no true and false, but popular and unpopular, and Rabbis are measured by their popularity and the number of their followers.

You ask, where does 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? Alas, my foray into Mi Yodea has failed to turn up any satisfactory answers (so far). Do you think it would be profitable to post your question there? Mind you, questions unrelated to Judaism, even if they are about... other religions, Hebrew language, Jews, Jewish history, and Israel ... are generally off-topic.

Please don’t shoot me down in flames for failing to answer your questions, because it’s not for the lack of trying. It’s also a very good question, which I up-voted, and I hope you will get more answers.

The idea that individual letters mean something is very old.

Judaism has always regarded Hebrew as a sacred language, the medium of divine communication. For millennia, its sages and mystics have taught that the letters are no ordinary expression. Indeed, the very word for “letter” in Hebrew–Ot–means sign or wonder; that is, a heavenly revelation. -source-

Here is one application thereof. Keep in mind it reads from right to left.

The first sentence of the Bible reads in Hebrew: בְּרֵאשִׁית בָּרָא אֱלֹהִים אֵת הַשָּׁמַיִם וְאֵת הָאָרֶץ B’raysheet ba-RA Eh-lo-HEEM ayt ha-sha-MA-yim v’-ayt ha-A-retz Ayt is made up of the first and last letters of the Hebrew alphabet, aleph/tav. In fact, the Hebrew alphabet is referred to as aleph/tav. Therefore, the literal translation of the first sentence in the Bible can be understood as, “In the beginning, God created the Hebrew alphabet and then He created the heavens and the earth.” From this, the sages understood that the the Hebrew alphabet was used as building blocks to create the heavens and the earth. -source-

That first letter, like a backwards english C, indicates to some that what came before is essentially none of our business, but what is thereafter may be.

Some associate this view with the Jewish Kabbalah. Some of these believe it began in Eden or from God to Moses. Typically it is based on an oral tradition.