σημεῖον sēmeîon - sign
This word is presumed to have derived from the word σημαίνω sēmaínō
meaning, a mark, to indicate or signify. When the angel of the Lord says: this will be a sign to you (NIV), we could ask ourselves the question, "what does this signify?" of the baby wrapped in cloths and lying in a manger. (NIV)
12 This will be a sign to you: You will find a baby wrapped in cloths and lying in a manger.”
New International Version (NIV)
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Contextually, verse 11 gives us two pieces of information: 1. a Saviour has been born, 2. He is the Messiah, the Christ.
11 Today in the town of David a Saviour has been born to you; he is the Messiah, the Lord.
So, the sign is the babe wrapped in cloths - there's a great discussion on this on another stack-exchange site. Here's the link: https://christianity.stackexchange.com/questions/12197/what-is-significant-about-the-swaddling-clothes-with-which-jesus-was-wrapped-as
And the sign points to the Saviour, Christ. Therefore Luke is not actually calling the birth of Christ a sign, but that the shepherds will find a babe wrapped in cloths, as a sign that the promised Messiah has come.