What is your opinion of giving Eidhiah (money) to children during Eid-ul-Fitr and Eid-ul-Adha? I was told that it is an innovation. Is this true?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Giving feast's gift is a good practice since it strengthens the kinship ties and develops love and affection among family members and Muslims in general.  However, the parents have to give such gifts to their children equally since dealing justly with children is an obligation.  Imams Al Bukhari and Muslim reported from Al Nuaman bin Bashir that he gave a gift to his son and went to the Prophet (Sallallahu Alaihi wa Sallam) to take his witness to that.  The Prophet (Sallallahu Alaihi wa Sallam) enquired him, "Have you given (a similar gift) to each one of your children?'  He replied in the negative.  Allah's Apostle said, "Fear Allah, and be just to your children."  As a result Al Nuaman took back the gift from his son. 

In a narration of Saheeh Muslim the Prophet (Sallallahu Alaihi wa Sallam) said to Al Nuaman.  "It is not correct, and I do not bear witness except in what is true."  A third narration reads, "I do not bear witness on injustices."

However, a father should know where his children spend the gifts, if they spend it in Haram way then giving gifts to them is not allowed.

Allah knows best.