Hi, Mira —
Thanks for the question.
It just so happens that the Catechism of the Catholic Church addresses your exact questions!
— What do the words of the Hail Mary prayer mean?
The Catechism tells us:
2673 In prayer the Holy Spirit unites us to the person of the only Son, in his glorified humanity, through which and in which, our filial prayer unites us in the Church with the Mother of Jesus.**
** Acts 1:14 They all joined together constantly in prayer, along with the women and Mary the mother of Jesus, and with his brothers.
So the words of the Hail Mary affirm the blessings the Lord had given to His own Mother and to us, in making His mother, also our mother.
It's important to keep in mind, we are not saying that Mary is the Mother of God the Father.
No, Mary is the mother of the Second Person of the Trinity, Jesus, who became incarnate (in the flesh) for our salvation!
Any Catholic who believes Mary is the Mother of God the Father is in heresy.
We would said Mary is:
- daughter of the Father
- mother of the Son, and
- spouse of the Holy Spirit
— More specifically, what does "the Lord is with thee" mean?
From CCC 2676 of the Catechism:
"Full of grace", "the Lord is with thee": These two phrases of the angel's greeting (Luke 1:28) shed light on one another. Mary is full of grace because the Lord is with her. The grace with which she is filled is the presence of him who is the source of all grace. "Rejoice . . . O Daughter of Jerusalem . . . the Lord your God is in your midst." (Zephaniah 3:14,17a) Mary, in whom the Lord himself has just made his dwelling, is the daughter of Zion in person, the ark of the covenant, the place where the glory of the Lord dwells. She is
"the dwelling of God . . . with men." (Revelation 21:3)
Full of grace, Mary is wholly given over to him who has come to dwell in her and whom she is about to give to the world.
— Does it just mean the Lord has found favor with Mary?
It means Mary has found favor with the Lord, thought what you have said is probably true as well.
Hope this answers your questions.
Mike and Eric