Was eternal life promised by the Mosaic Law? John 3:16 (NIV) 16 For God so loved the world that he gave his one and
  only Son, that whoever believes in him shall not perish but have
  eternal life.


The New Testament clearly tells us how one can attain salvation or eternal life.

What about Old Testament? Was there any promise of eternal life associated with any of the Mosaic Law. For instance, was animal sacrifice promised with eternal life?

How can we understand this verse?


  Matthew 19:16 (NIV)
  Just then a man came up to Jesus and asked, “Teacher, what good thing
  must I do to get eternal life?”


If the rich man was a Jew, he must have known how to attain eternal life from the Torah.
Was there any promise of eternal life associated with any of the
  Mosaic Law. For instance, was animal sacrifice promised with eternal
  life?


No.  Eternal life is foretold in the Old Testament, but never as a promise associated to obedience to Mosaic Law.

The following verses refer to life after death, hinting at the eternal nature of life.   This explains the how the man in Matthew 19:16 knew about eternal life. 

All are NIV. 


  Psalm 71:20
  
  Though you have made me see troubles, many and bitter, you will
  restore my life again; from the depths of the earth you will again
  bring me up.
  
  Isaiah 26:19 
  
  But your dead will live; their bodies will rise. You who dwell in the
  dust, wake up and shout for joy. Your dew is like the dew of the
  morning; the earth will give birth to her dead.
  
  Ezekiel 37:12-14
  
  12 Therefore prophesy and say to them: ‘This is what the Sovereign
  Lord says: My people, I am going to open your graves and bring you up
  from them; I will bring you back to the land of Israel. 13 Then you,
  my people, will know that I am the Lord, when I open your graves and
  bring you up from them. 14 I will put my Spirit in you and you will
  live, and I will settle you in your own land. Then you will know that
  I the Lord have spoken, and I have done it, declares the Lord.’”


And one that clearly teaches everlasting life as part and parcel with resurrection:


  Daniel 12:2
  
  Multitudes who sleep in the dust of the earth will awake: some to
  everlasting life, others to shame and everlasting contempt.


However, these are prophetic or poetic works, not books of the Law.  I'm not aware of any teachings on eternal life in the Pentateuch - this first five books, which are commonly referred to as the "books of the Law".  Technically, those are "The Law of Moses" or "Mosaic Law", so it would seem that the answer to your question is "no" if you wish to be that precise, but there are Old Testament teachings/revelations/writings on it.

From a traditional Sola Fide perspective, you wouldn't expect it to, though. 

Most of the Mosaic laws talked about consequences or punishment for sin, along with a few promises of reward for obedience. 

However, Scripture clearly teaches that eternal life is not earned through obedience to the Law.   It is a gift from God, not of works.  (Ephesians 2:9.) We are not justified by works of the law (Galatians 2:16)

If the Mosaic Law promised eternal life based on obedience to the Law, the entire message of Christianity would be invalidated.