What is the essential argument in favor of the tripartite nature of man? A common belief regarding the nature of mankind is that we have a body, a soul and a spirit.  This is known as the Tripartite Nature of Man.  

What is the essential argument in favor of this view?  I would also be interested to know how early in the Scriptures this is indicated or understood.  It would be really interesting to know how the Old Testament speaks of this and whether or not that idea was held by Jewish people prior to the coming of Jesus.
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.
  
  
    
    a phantom, i.e. (figuratively) illusion, resemblance
    (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.
  
  
    
    resemblance
    (concretely) model, shape
    (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.
  
  
    
    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.
  
  
    
    alive
    (hence) raw (flesh)
    fresh (plant, water, year), strong
    (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.
  
  
    
    a current of air, i.e. breath (blast) or a breeze
    (by analogy or figuratively) a spirit
    (humanly) the rational soul
    (by implication) vital principle, mental disposition, etc.
    (superhumanly) an angel, demon
    (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.