"Hail Mary" is a reference to the prayer that Elise copied down. It can be used to refer directly to the prayer - for example, "I said ten 'Hail Mary's last night" means that someone prayed by reciting the prayer ten times. In the US, it is also used as an adjective to describe an action that you hope will succeed, but has a low chance of going well (usually due to lack of preparation, resources, and planning). The implication is that the only thing that will make the attempt succeed is prayer and divine intervention. It is often used in sports for a last, desperate attempt to win a game.

The "Hail Mary" is a Catholic prayer. Here is the full prayer: Hail Mary, full of Grace, the Lord is with thee. Blessed are thou among women and blessed is the fruit of thy womb Jesus. Holy Mary, Mother of God, pray for us sinners now and at the hour of our death. I hope that helps!