The foremost resistance to Marian prayer, or to praying to the Saints in general, is the idea of our prayers being directed to someone other than God. I understand this hesitancy since, for many, prayer is only a Godward activity. And while I agree that prayer is primarily a Godward activity, the Church correctly views prayer as entreaty. And one can entreat both God and man.
Others can and do intercede for us. Indeed, Samuel says, "far be it from me that I should sin against the Lord by ceasing to pray for you" (1 Sam 12.23). The Lord's Prayer is a prayer of and for community - a prayer of and for "us." And so we go to our mother and father, brother and sister, friend and neighbor, and pastor and ask them to pray for us. We ask them to intercede for us. Is their mediation on our behalf, is our request for their mediation, sinful or idolatrous? Of course not. Does our request for their intercession preclude our going also directly to Christ? Of course not.
And so it is with the Blessed Virgin Mary, the Apostles, and the Saints - they are not dead, but gloriously alive. They are not deaf and dumb and lame, but gloriously aware and active. We believe in the communion of saints: past and present, militant and victorious, on earth and in heaven. We believe they can and do pray for us and because of this, we ask it of them.
A common objection to prayers to Saints is our inability to figure out the How of it - How do they hear our prayers along with millions of others? How do they manage? And my answer to that question is that I don't know. But I also do not know what it means to be glorified, to be holy, to perfectly partake of the divine nature. Yet it is so. And the Church has always professed it to be so. I have no reason to doubt her authority, who also, carried along by the Holy Spirit, gave me the Scriptures.
Mary holds a special place for the Church because she is our Mother - Christ gave her to us and us to her at the cross (John 19.26,27). She is also the Mother of God, the Theotokos (God-bearer). This title does not imply that she is the Mother of the Father, but as my five year old, Anna, said, "Mary gave Jesus a birthday." The simplicity and profundity of Anna's statement is spot on. She is the Mother of God because Jesus is God the Son, and her Yes allowed Him to be born. His humanity is from her, just as mine is from my parents.
And so we pray to Mary, though, of course, not exclusively. She is not the mediator between God and Man - Christ is. But she mediates for us, in an intercessory sense, just as my earthly mother mediates for me, but more greatly and more profoundly. She acts as she acted at the wedding at Cana, bringing our needs (They have no wine) before Jesus, and instructing us, the Church (Do whatever He tells you).
I pray to God, of course - daily, hourly, moment by moment. And within the Rosary itself we praise the Trinity, pray to the Father, and cry out to Jesus.