St. Augustine says of the Blessed Virgin Mary (whose soul magnifies the Lord):
What the Lord magnified in her was, that she did the will of the Father, not that flesh gave birth to flesh. St. Augustine of Hippo - Tractate 10
That's what Mary magnified, as a handmaid, her soul magnifies the Lord. Augustine goes on to talk about:
May they ever say, “Exalted be the LORD who delights in the peace of his loyal servant.” (Psalm 35:27b NABRE)
which in his translation must have clicked with the Magnificat (Luke 1:46-55) because his says:
The Lord be magnified, they that wish the peace of His servant
So, exalting (or lifting the name of) God, it's an action of the soul, which we desire God to do to us. Mary's "Yes" meant something, it meant that she desired that her soul would be open to accept the Holy Spirit. While at the same time she, like John, must decrease as well as suffer, so her Son could increase.
Depending on your understanding or opinions regarding the faculties of the soul it would seem as though there's not a lot a person can do to influence one's soul. A devout prayer that God should allow a soul to magnify Him (and this is a prayer, in the Psalms and the Gospel) is the an excellent way, an perhaps one of the only ways, to open ourselves up to God's grace.