The breath of life was breathed into Adam was his own individualized spirit.
The part of Adam that had existed before he was created and continued to exist after he died.
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We all have such spirit that existed before our birth and will continue to exist after our death.
It is this spirit that contains our persona or individuality. It is this spirit that we seek to improve upon and grow and that this mortal phase of our existence is designed to test and try.

It is not mentioned often in the scriptures but it's there. Besides the reference in connection with Adam in Genesis we find it mentioned in more places in both the Old and New Testaments. It is separate from the flesh and is sometimes referred to as the soul; or alternatively the spirit and the flesh combined can be refereed to as the soul.

for example from the Old Testament:
Numbers 16:22 And they fell upon their faces, and said, O God, the God of the spirits of all flesh, shall one man sin, and wilt thou be wroth with all the congregation?

Numbers 27:16 Let the Lord, the God of the spirits of all flesh, set a man over the congregation,

1 Kings 17:20 And he cried unto the Lord, and said, O Lord my God, hast thou also brought evil upon the widow with whom I sojourn, by slaying her son?
21 And he stretched himself upon the child three times, and cried unto the Lord, and said, O Lord my God, I pray thee, let this child’s soul come into him again.

Job 32:8 But there is a spirit in man: and the inspiration of the Almighty giveth them understanding.

Ecclesiastes 12:7 Then shall the dust return to the earth as it was: and the spirit shall return unto God who gave it.

from the New Testament:
Matthew 26:40 And he cometh unto the disciples, and findeth them asleep, and saith unto Peter, What, could ye not watch with me one hour?
41 Watch and pray, that ye enter not into temptation: the spirit indeed is willing, but the flesh is weak.

James 2:26 For as the body without the spirit is dead, so faith without works is dead also.

Hebrews 12:9 Furthermore we have had fathers of our flesh which corrected us, and we gave them reverence: shall we not much rather be in subjection untothe Father of spirits, and live?

1 Peter 3:18 For Christ also hath once suffered for sins, the just for the unjust, that he might bring us to God, being put to death in the flesh, but quickened by the Spirit:
19 By which also he went and preached unto the spirits in prison;
20 Which sometime were disobedient, when once the longsuffering of God waited in the days of Noah, while the ark was a preparing, wherein few, that is, eight souls were saved by water.

It is this spirit within each of us, that was created by God, that makes Him literally our Father in Heaven and each one of us his children.

Deuteronomy 14:1 Ye are the children of the Lord your God: ye shall not cut yourselves, nor make any baldness between your eyes for the dead.
2 For thou art an holy people unto the Lord thy God, and the Lord hath chosen thee to be a peculiar people unto himself, above all the nations that are upon the earth.

Job 33:4 The Spirit of God hath made me, and the breath of the Almighty hath given me life.

Psalms 82:6 I have said, Ye are gods; and all of you are children of the most High.

Isaiah 42:5

Thus saith God the Lord, he that created the heavens, and stretched them out; he that spread forth the earth, and that which cometh out of it; he that giveth breath unto the people upon it, and spirit to them that walk therein:

Hosea 1:10

Yet the number of the children of Israel shall be as the sand of the sea, which cannot be measured nor numbered; and it shall come to pass, that in the place where it was said unto them, Ye are not my people, there it shall be said unto them, Ye are the sons of the living God.

Zechariah 12:1

The burden of the word of the Lord for Israel, saith the Lord, which stretcheth forth the heavens, and layeth the foundation of the earth, and formeth the spirit of man within him.

Malachi 2:10

Have we not all one father? hath not one God created us? why do we deal treacherously every man against his brother, by profaning the covenant of our fathers?

Acts 17:28

For in him we live, and move, and have our being; as certain also of your own poets have said, For we are also his offspring.
29 Forasmuch then as we are the offspring of God, we ought not to think that the Godhead is like unto gold, or silver, or stone, graven by art and man’s device.

Romans 8:16

The Spirit itself beareth witness with our spirit, that we are the children of God:

Ephesians 4:5

One Lord, one faith, one baptism,
6 One God and Father of all, who is above all, and through all, and in you all.

Hebrews 12:9

Furthermore we have had fathers of our flesh which corrected us, and we gave them reverence: shall we not much rather be in subjection untothe Father of spirits, and live?

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One of the clearest verses on our pre-mortal existence as spirits is where the Lord speaks to Jeremiah in this manner.
Jeremiah 1:4 Then the word of the Lord came unto me, saying,
5 Before I formed thee in the belly I knew thee; and before thou camest forth out of the womb I sanctified thee, and I ordained thee a prophet unto the nations.

Paul mentions that many of us were chosen in this spiritual state.
Ephesians 1:3 > Blessed be the God and Father of our Lord Jesus Christ, who hath blessed us with all spiritual blessings in heavenly places in Christ:
4 According as he hath chosen us in him before the foundation of the world, that we should be holy and without blame before him in love:

Similarly, Adam had been chosen and ordained to be the first of these spirit children of God to be "breathed into" and inhabit a physical body on this earth.
He too was chosen with his wife eve to open the path that all the other spirit children of God would have the same privilege.

Eden, the two trees and the opportunity to make a choice and exercise free will, were key to this process and opening of the way for all the spirit sons and daughters of God our literal Father in Heaven.