The Hebrew Bible speaks of "new birthing" involving water, which then later carries into the Christian New Testament. Specifically, water in the Hebrew Bible removes the contamination of death, and thus emerges "new life."

To begin, the simple Hebrew verb [**חָטָא**][1] means *to sin*, but in the intensive forms of the verb (Piel, Pual, and Hithpael) the meaning includes to "de-sin," and therefore to cleanse from the contamination of death.  In this sense, the intensive idea appears in the Hebrew Bible with water, which will "de-sin" the object contaminated by death (or dead flesh, which is leprosy). In the following two verses, the Hebrew verb [**חָטָא**][2] occurs three times in the third person masculine singular Hithpael (imperfect) form.

>[**Numbers 19:12-13**][3] (MT)   
הוּא **יִתְחַטָּא**-בוֹ בַּיּוֹם הַשְּׁלִישִׁי, וּבַיּוֹם הַשְּׁבִיעִי--יִטְהָר; וְאִם-לֹא **יִתְחַטָּא** בַּיּוֹם הַשְּׁלִישִׁי, וּבַיּוֹם הַשְּׁבִיעִי--לֹא יִטְהָר  
 
><sup>12</sup> **That one shall purify himself** from uncleanness with the water on the third day and on the seventh day, and then he will be clean; but if **he does** not **purify himself** on the third day and on the seventh day, he will not be clean. 

>כָּל-הַנֹּגֵעַ בְּמֵת בְּנֶפֶשׁ הָאָדָם אֲשֶׁר-יָמוּת וְלֹא **יִתְחַטָּא**, אֶת-מִשְׁכַּן יְהוָה טִמֵּא--וְנִכְרְתָה הַנֶּפֶשׁ הַהִוא, מִיִּשְׂרָאֵל:  כִּי מֵי נִדָּה לֹא-זֹרַק עָלָיו, טָמֵא יִהְיֶה--עוֹד, טֻמְאָתוֹ בוֹ

><sup>13</sup> Anyone who touches a corpse, the body of a man who has died, and **does** not **purify himself**, defiles the tabernacle of the Lord; and that person shall be cut off from Israel. Because the water for impurity was not sprinkled on him, he shall be unclean; his uncleanness is still on him.

The Hithpael is reflexive, so the subject of the verb acts on himself. The exact same verb form occurs again in [Numbers 8:21][4], [Numbers 19:20][5] and [Numbers 31:23][6] within the same context of ritual self-purification within the Mosaic Law. So anyone who was contaminated by dead people or animals (or had been contaminated with dead skin, which was leprosy in [Leviticus 14:1-57][7]) were made clean with the water flowing under sacrificed animals and/or filtered through ashes of the red heifer, and then sprinkled by means of some red fabric/cedar/hyssop branch.

In summary, the removal of the contamination of death occurred through washing with water. Death is "washed away" with water.

Finally, in the Book of Ezekiel the major promise of washing by water occurs; that is, the "New Covenant" would be inaugurated with the cleansing of water.

>[**Ezek 36:24-26**][8] (NASB)   
<sup>24</sup> For I will take you from the nations, gather you from all the lands and bring you into your own land. <sup>25</sup> **Then I will sprinkle clean water on you, and you will be clean; I will cleanse you from all your filthiness and from all your idols.** <sup>26</sup> Moreover, I will give you a new heart and put a new spirit within you; and I will remove the heart of stone from your flesh and give you a heart of flesh.

In the Book of Ezekiel, idolatry was connected with dead people, and therefore the House of Israel was contaminated with death, and so needed this washing. 

>[**Ezekiel 6:4-5**][9] (NASB)   
<sup>4</sup> So your altars will become desolate and your incense altars will be smashed; and I will make your slain fall in front of your idols. <sup>5</sup> I will also lay the dead bodies of the sons of Israel in front of their idols; and I will scatter your bones around your altars.

In the Christian New Testament, which is the "New Covenant" interpretation by Christians, the idea carries over: the blood of Jesus atones for sin, but his eternal life washes away the spiritual death of man as promised by Ezekiel, because his eternal life is "living water." That is, the death of Jesus was the sin/guilt offering, and his water (which was "filtered through" His sacrifice) is his eternal life, which washes away the contamination of death so that we may have life eternal.

The following verses draw this distinction.

>[**Romans 5:8-11**][10] (NASB)   
<sup>8</sup> But God demonstrates His own love toward us, in that while we were yet sinners, Christ died for us. <sup>9</sup> Much more then, having now been justified by His blood, we shall be saved from the wrath of God through Him. <sup>10</sup> For if while we were enemies we were reconciled to God through the death of His Son, much more, having been reconciled, we shall be saved by His life. <sup>11</sup> And not only this, but we also exult in God through our Lord Jesus Christ, through whom we have now received the reconciliation.

The sinner is "justified" by the blood (sin/guilt atonement), but is only reconciled or "saved" through his eternal life (washing away of spiritual death).  The Holy Spirit applies the "living water" of Christ to the sinner, and thus the Holy Spirit "baptizes" the believer into the body of Christ. Thus the believer is born "by spirit and water" ([Jn 3:5][11]).

Thus while Jesus died for all men ([1 Jn 2:2][12]), only those who receive eternal life are saved. 

In summary, the idea of the new birth stemmed from the Hebrew Bible, where not blood, but water was used to remove the contamination of death. When this contamination is removed, such a person is now alive (because death was washed away). This now-being-alive is the new birth, which Jesus, Paul, and Peter describe as born again. 

  [1]: http://www.biblestudytools.com/lexicons/hebrew/nas/chata.html#Legend
  [2]: http://www.biblestudytools.com/lexicons/hebrew/nas/chata.html#Legend
  [3]: http://www.mechon-mamre.org/p/pt/pt0419.htm
  [4]: http://www.biblegateway.com/passage/?search=Numbers%208:21&version=NASB
  [5]: http://www.biblegateway.com/passage/?search=Numbers%2019:20%20&version=NASB
  [6]: http://www.biblegateway.com/passage/?search=Numbers%2031:23&version=NASB
  [7]: http://www.biblegateway.com/passage/?search=Leviticus%2014:1-57&version=NASB
  [8]: http://www.biblegateway.com/passage/?search=Ezek%2036:24-26&version=NASB
  [9]: http://www.biblegateway.com/passage/?search=Ezekiel%206:4-5&version=NASB
  [10]: http://www.biblegateway.com/passage/?search=Romans%205:8-11&version=NASB
  [11]: http://www.biblegateway.com/passage/?search=John%203:5&version=NASB
  [12]: http://www.biblegateway.com/passage/?search=1%20Jn%202:2&version=NASB