I find this sort of stuff fascinating, so I highly recommend the Wikipedia page on the Names of God. Quoting from that page:

YHWH is an archaic third person singular imperfect of the verb “to be” (meaning, therefore, “He is”). This interpretation agrees with the meaning of the name given in Exodus 3:14, where God is represented as speaking, and hence as using the first person (“I am”).

So “He who is being,” is the way that the Israelites named God – NB, not “He who is already complete:” YHWH is imperfect, and so denotes an activity that is presently underway (so, for that matter, does the English “is;” when we say of a hammer that “it is,” we mean that “it is being,” “its being is now underway;” a being that is finished with being no longer is, but rather was, and is no more). When God named himself, he called himself – again, using an imperfect form of “to be” – “I who am being I:”

Ehyeh asher ehyeh (Hebrew: אהיה אשר אהיה) is the first of three responses claimed to be given to Moses when he asks for God’s name (Exodus 3:14). The Tetragrammaton itself derives from the same verbal root. The King James version of the Bible translates the Hebrew as “I Am that I Am” and uses it as a proper name for God.

Ehyeh is the first-person singular imperfect form of hayah, “to be.” Ehyeh is usually translated “I will be,” since the imperfect tense in Hebrew denotes actions that are not yet completed (e.g. Exodus 3:12, “Certainly I will be [ehyeh] with thee.”). Asher is an ambiguous pronoun which can mean, depending on context, “that”, “who”, “which”, or “where.”

Although Ehyeh asher ehyeh is generally rendered in English “I am that I am,” better renderings might be “I will be what I will be” or “I will be who I will be,” or “I shall prove to be whatsoever I shall prove to be” or even “I will be because I will be.” In these renderings, the phrase becomes an open-ended gloss on God’s promise in Exodus 3:12. Greek, Ego eimi ho on (ἐγώ εἰμι ὁ ὤν), “I am The Being” in the Septuagint, and Philo, and Revelation or, “I am The Existing One”; Latin, ego sum qui sum, “I am Who I am.”

What was the glyph that the Hebrews used to indicate the Tetragrammaton (when, e.g., they were anointing kings, priests or prophets)? It was the paleo-Hebrew letter taw, which looks like a tilted x; the same alphabetic root gave us the letter t.

This is the same glyph that Christian priests still use to anoint royal prophets of the priestly order of Melchisedek after baptizing them in the Name of the Tetragrammaton. The Cross as a symbol of the LORD, and of the Ground of all Being, and of the fundamental existential act, goes back a long, long way before the Cartesian coordinate system or the chi-rho, or the crucifixion.

As to whether YHWH refers to the Father, Son or Holy Ghost, the answer, as Alan says, is “yes.” Each person of the Trinity suffuses the others – we could equivalently say that they know and reflect each other perfectly, and participate in each other thoroughly. So, where one operates, all three operate. This is true both in the internal economy of the Trinity, and in its outward operations.

Nevertheless YHWH was taken by the ancient Israelites to refer primarily to the Son, the King of Heaven, Lord of his Father El’s household, Captain of the Heavenly Host of the angels, and President and God of the council of the gods, or angels, who were created by El rather than begotten of him, and then adopted as his sons. El was the Most High God of the ancient Semitic pantheons. Each of his sons was a ba’al – the ancient Semitic term for the Lord of a Household under a patriarch, literally “bar-El.” Each nation had a divine Shepherd King, its own god, its own ba’al, appointed to rule over it by El; the king of a city was generally understood as the incarnation of its ba’al (raising a clear requirement for the possibility that the incarnate ba’al might die and rise again: the King is dead, Long Live the King). The ba’al of Israel was YHWH, a storm god who seems first to have had his mountain house somewhere south of Israel. All the ba’alim had mountain houses, or gardens, like Olympos. Eden was one such, Sinai another, and Zion a third. When you read of “high places” in the Bible, you are reading of temples to ba’alim erected upon their mountains (so when you hear in the Psalms that YHWH made the mountains skip like rams, you may understand this to mean that he made the ba’alim jump at his command).

One of the things Israel figured out over the course of her history was that YHWH was the only begotten son of El, and thus more powerful than all other gods (this being the lesson that Elijah taught to the prophets of ba’al, and so to the Israelites). In other words, she figured out that YHWH was the heir of the Most High God, who had created all things. And as she figured this out, or rather as it was revealed to her, she more and more treated YHWH and El as if they were effectually the same being. Which, of course, they are.

As to the analogy from Mind of the Maker that Bedarz mentions, it seems OK to me (bearing in mind that all I know of it is what Bedarz has just said), provided we remember that the whole Trinity is prior to time, and also that the whole Trinity is manifest in time (as it is manifest everywhere and everywhen)(this being but a different way of saying that God is necessary). Finally, we ought to remember that it is the whole Trinity that is necessarily manifest, first prior to all worlds, and second in all worlds, which are as it were the fossils, relicts and artifacts of the Divine Act, engraved images or projections of his being.

I would close by affirming what Alan has said about these intellectual constructs, ancient and modern: they must be taken only as ways of understanding the reality toward which they point, and as indications of its character, rather than as in any way dispositive in their own right. They are images and icons, at the very best. They are not to be confused with the real being they indicate and elucidate. We ought not to bear them in mind as we worship, or pray. They are nothing more than ways to settle difficulties we are worried about, so that we can approach the altar with minds free of any encumbrances of doubt, and clear of confusion. They should be set aside, together with our sins, before we turn toward the Presence and call upon the Name.

As for the analogy you present, it has some validity, but we must ultimately trust Scripture to tell us about God, and not make too much of the intellectual constructs we use.

Also, what do you think of the analogy presented in The Mind of the Maker? Father as the Creative idea (timeless) , the Son as the Energy or Activity (that necessarily manifests itself in space-time and the Ghost as the Power of the creativity?