What is the essential argument in favor of the tripartite nature of man?

I have found the 1 Thess.5:23 passage's inclusion of "spirit and soul and body" very compelling in that it affirms a nature given to man, one of bearing the image of God, that is easily associated with the knowable nature of God (as opposed to all that which is unknowable), His triune existence.
There seems to be an example of this type of unity that may serve to illustrate it: the computer. A computer exists in three senses. It has a body, all that is material. It has an operating system which serves as its basis of operation, which parallels the 'spirit'. Its first two components then form the vehicle through which software can do useful functions, which parallels the 'soul'. In this illustration, the 'spirit', or system, is the repository of the values, or things that are permissible for the computer to do. Values are indeed maintained here in the 'heart' of the computer. (In the computer systems that are available to purchasers, there is a choice between two systems: Apple or PC. There are two options of internal values in humans which are manifested: one in the lost and one in the redeemed.) Also, in this illustration, the 'soul' (software/applications) is dedicated to doing its work as it is designed to be in accord with what is permissible by its operating system (spirit); i.e. Apple programs won't run on a PC. No illustration is perfect in such matters as these, but in this there might be something of value. The heart, or spirit, contains our values (either God-placed or of ungodly origin). When the soul is about its daily business of directing the body, there is constant communication with the spirit, consulting over what those values permit or disallow. There seems to be a working interface between these separate entities that would be called 'the conscience'. When God vacated man's spirit, this place of values became open to all other input. In the unregenerate, there may be remnants of godly values mixed with much of the ungodly. In the regenerate, the spirit becomes the place of the Holy Spirit's residence and begins the patient, internal work of forming God's values there.

In addition to man being made in the image of the Triune God as BYE has argued (and is therefore analogously also a triune being), those supporting the tripartite view of man's nature also argue that the biblical teaching of the believer being a temple of the Holy Spirit (cf. 1 Corinthians 6:19) implies trichotomism:

The Bible compares man to the temple. In doing so it depicts the three elements of man most conspicuously.

We know that the temple is divided into three parts. The first part is the outer court, which everyone can see and into which everyone can go. All outward worship is offered to God here. Further in, there is the Holy Place. Here, only the priests can enter. In the Holy Place the priests offer the blood, the oil, the incense, and the bread to God. Although they are very close, they are not the closest, because they are still outside the veil and cannot enter into God's presence. God dwells in the Holy of Holies from which He radiates His infinite glory. The Holy of Holies is otherwise dark. No one can come before Him. Although the high priest is allowed once a year to enter the Holy of Holies, this only demonstrates all the more that before the rending of the veil there was no one within the veil.

Man is a temple of God. Within man there are also three parts. The body is like the outer court; it is outside, and its life is seen by all. It is here that man should obey all God's commandments. It is also here that God's Son died for man. Further in, there is man's soul, which is the inward life in man; it includes man's feelings, will, and mind. This is the Holy Place to a regenerated one. His love, thoughts, and desires are all here. In this place there is much light, everything is clear and obvious, and the priests come in and out to serve God. However, further in, there is the Holy of Holies behind the veil, which is unreachable by human light and is a place invisible to the human eyes. This is "the secret place of the Most High" (Psa. 91:1). It is the habitation of God, a place that no man can reach unless God removes the veil. This is the human spirit. Man not only has a body and a soul but a spirit as well. This spirit is deeper than man's consciousness; it is the place unreachable by man's feelings. It is in this place that man fellowships with God. - The Spiritual Man, Watchman Nee

Simeon has already mentioned 1 Thessalonians 5:23 -

May God himself, the God of peace, sanctify you through and through. May your whole spirit, soul and body be kept blameless at the coming of our Lord Jesus Christ. (NIV)

But I believe an equally strong if not stronger proof text is found in Hebrews 4:12 -

For the word of God is alive and active. Sharper than any double-edged sword, it penetrates even to dividing soul and spirit, joints and marrow; it judges the thoughts and attitudes of the heart. (NIV, emphasis added)

The implication here is that the soul can be distinguished from the spirit as two separable entities maintained within a thinking (and therefore living) being. On the surface at least, this seems to be incompatible with a dichotomous view.

Your question is one with which I wrestled for several years before making these discoveries in Scripture and I hope that it will give you the answer you are seeking.

Genesis 1:26 and 27 King James Version

26 And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth.

27 So God created man in his own image, in the image of God created he him; male and female created he them.

So I researched the word image and it's true meaning. The original Hebrew word was:

צֶּלֶם

tselem (tseh'-lem) n-m.

1. a phantom, i.e. (figuratively) illusion, resemblance

2. (hence) a representative figure, especially an idol [from an unused root meaning to shade]

KJV: image, vain shew.

notice the translation Phantom. this word has no connection to any physical substance, and in this case must be talking about other properties other than Physical. That leaves us to deduce that it must refer to spiritual characteristics.

Also please note the word * likeness* in verse 26. the original word used was:

dmuwth (dem-ooth') n-f.

1. resemblance

2. (concretely) model, shape

3. (adverbially) like

KJV: fashion, like (-ness, as), manner, similitude.

Since the word is used in its adverbial form it means like, and the word our makes it plural.

Actually in the interlinear Hebrew translation the phrase 'in our image' was:

בְּצַלְמֵ֖נוּ

bə•ṣal•mê•nū

which literally translated means 'in our image.'

This leaves us to understand that it is in reference to Spiritual beings and it must refer to a multiplicity.

So how do we get from there to what the likeness to that multiplicity means regarding man.

We must go back to: Genesis 2:7 KJV

And the LORD God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.

We need to begin with what is the breath of life.

Breath: נְשָׁמָה

nshamah (nesh-aw-maw') n-f.

1. a puff, i.e. wind, angry or vital breath, divine inspiration, intellect. or (concretely) an animal

KJV: blast, (that) breath(-eth), inspiration, soul, spirit.

Life: חַי

chay (khah'-ee) adj.

1. alive

2. (hence) raw (flesh)

3. fresh (plant, water, year), strong

4. (as noun, especially in the feminine singular and masculine plural) life (or living thing), whether literally or figuratively

KJV: + age, alive, appetite, (wild) beast, company, congregation, life(-time), live(-ly), living (creature, thing), maintenance, + merry, multitude, + (be) old, quick, raw, running, springing, troop.

It is imperative that we first understand that the noun here is nshamah or breath and chay alive is an adjective.

That 'God breathed into his nostrils' means that God transferred from himself to man's nose his breath. The noteworthy point here is that since God is Eternal that of necessity requires that his breath is also eternal.

Why this is important lies in what is said next ' man became a living Soul.

The significance of that is that the Soul created by the breath of life is an eternal being.

The importance of that was given by Christ in:

Matthew 10:28 KJV

And fear not them which kill the body, but are not able to kill the soul: but rather fear him which is able to destroy both soul and body in hell.

The true significance of that statement is that only God has the ability to destroy the soul.

So now we have covered the body and soul, but if we are to be in the image of the trinity we need a third part, and that third part is the Spirit.

So what is the Spirit, and how is it different from the soul? The Hebrew word is:

πνεῦμα

pneuma (pnyoo`-mah) n.

1. a current of air, i.e. breath (blast) or a breeze

2. (by analogy or figuratively) a spirit

3. (humanly) the rational soul

4. (by implication) vital principle, mental disposition, etc.

5. (superhumanly) an angel, demon

6. (divinely) God, Christ's spirit, the Holy Spirit

KJV: ghost, life, spirit(-ual, -ually), mind

According to the Hebrew it is the rational soul, or the vital principle, or mental disposition. So the Spirit and soul are inextricably joined and the Spirit is the composite of all the persons knowledge and personality.

These are the conclusions I have reached from my study of the Bible.