But it is in the Bible. When the angel Gabriel was sent to Mary by the Father, he greeted her, “Hail, full of grace; the Lord is with you” (Luke 1:28). When Mary visited Elizabeth, Elizabeth exclaimed, “Of all women you are the most blessed, and blessed is the fruit of your womb” (Luke 1:41-42). So here we have the first part of the Hail Mary. St. Bernardine added the name of Jesus, clarifying who the fruit of her womb is.
The second part is in response to the first, asking Mary to pray for us and addressing her as the mother of God. Of course, Elizabeth addressed her in much the same way: “Why should I be honored with a visit from the mother of my Lord?” (Luke 1:43).
Since Gabriel’s words were at the request of God, those words were actually God’s words. So we are honored to repeat them.
For more on praying to Mary, see Praying to Mary – A Biblical Defense.