Here are some examples. Eight days into his life his parents had him circumcised and named according to the Jewish Law. (Luke 2:21; Leviticus 12:3) Forty days after his birth, his mother and Joseph traveled to Jerusalem to complete God's instructions for purification and presentation of a male son to God. (Luke 2:22-24; Leviticus 12) Luke also tells us that the family made the annual trip to Jerusalem for the Festival of Passover. (Luke 2:41; Exodus 12; which we know mostly by the Seder.)
What was unique about Jesus' early life were the people's unsolicited confession as to who he was. (Read Luke 2:22-52 to read some of those.) In the middle of his parent's faithful observance of God's Law God revealed who this special child was and was to become.
So, what about Hanukkah? We are not told in the New Testament that Jesus or his family actually practiced Hanukkah, but the festival was in place in his day and given his family's loyalty to the traditions of Israel, we can suppose Jesus practiced the observance in its first-century form. The Gospel of John noted that Jesus was in Jerusalem in winter at the time of Hanukkah, known as the Festival of Dedication. (John 10:22, 23)
What's my point? My point is that Jesus learned from his parents to observe the laws and traditions God gave Israel, and every indication is that he observed them into his adult life. What is different now is that Jesus did for the festivals what he did for the other traditions and laws: He fulfilled them.
When Jesus stepped out on the public stage, he declared he had not come to destroy God's law, but he had come to complete it, fulfill it, show us the motives behind the practices in order to have a right-filled relationship with God. (Matthew 5:17-20) Jesus respected and observed the laws God gave His people--but his mission was to complete them. This is why his followers don't observe those laws and traditions today. He finished them. Jesus is our worship.
Hanukkah is not one of the festivals God prescribed for Israel, but it is part of Jewish tradition that celebrates God's deliverance and God's provision. Jesus knew of and possibly observed that tradition, and he is God's final deliverance and provision of all people.
We who celebrate Jesus through the festival of the Christ Mass, which God did not prescribe either, must also remember and worship the Person of God's deliverance and Provision, not the traditions we have built up around those truths.
May this season of lights point to the true Light, Jesus. (John 8:12)