Is there any Biblical support for the belief of Catholics [and others] to call upon Mary to intercede to God on their behalf?
First of all, because Mary was the birth mother of the humanity of Jesus in who "the fullness of the deity (God) was pleased to dwell bodily," [Col. 1:19, 2:9] Catholics [and others] nize the Virgin Mary as the Mother of God.
Secondly, because Mary was "favoured by God" [Lk. 1:30] when she was personally chosen by the Lord to become the mother of Jesus, God incarnated, Catholics [and others] believe that they have a greater chance of obtaining God's grace for their daily physical and spiritual needs by asking Mary to intercede on their behalf.
Thirdly, the above mentioned belief is partially based on the fact that Jesus is the King of kings and Mary, as the mother of the King, is the "queen mother." Then, when studying Jewish history, it is discovered that the institution of the "queen mother" was established during the reign of King Solomon.
In the Old Testament, we learn of the favoured position of the queen mother through the following words, "... then the king sat on his throne, and had a throne brought for the king's mother, and she sat on his right. Then she said, 'I have one small request to make of you, do not refuse me.' And the king said to her, 'Make your request, my mother, for I will not refuse you.'" [1 Kgs. 2:19-20]
Following the reign of King Solomon, many of the kings kept this practice. The mother of the king, through who the king received his throne, was trusted as a confident and advisor. So important was the function of the queen mothers in the days of the Old Testament that their names were listed in the succession records of the kings of Judah. [See 1 Kgs 14:21, 15:13; 2 Kgs. 12:1, 14:2, 15:2, etc...]
Similar to the intercession of the Queen Mother, when a child desires a favour and cannot obtain it from his/her parents, frequently the child will make the request to the grandparent to intercede on his/her behalf, therefore obtaining the favour that was being sought. This does not mean that the child is seeking the parental favour from the grandparent. Rather, the child is seeking the intercession of the grandparents before the parents.
Furthermore, Catholics [and others] don't differentiate between the living (in this world) and the dead (those who departed) members of the Body of Christ (the fullness of the Church.) The fullness of the mystical Body of Christ is found in the union of all the saints, past and present, here below and those above in Heaven.
In view of the aforementioned, it cannot be denied that those who were called to Heaven, including Mary, are still alive in spirit in the Kingdom of God that coexists with our world. As some non-Catholics and other non-liturgically based regious people pray through their deceased parents, grandparents or other biological relatives, asking these beloved departed persons to intercede before God on their behalf, Catholics [and others] pray through Mary to Jesus, taking advantage of her blessed position as the Mother of God.
This action affirms the prophetic and Divinely inspired passage that is found in the Holy Bible where it states, "Surely, from now on all generations will call me blessed." [Lk. 1:47] All generations couldn't call Mary blessed if she was not actively involved in the progressive Divine Plan that continues to develop before our eyes.
Indeed, all generations have called the Blessed Virgin Mary blessed, including this one, because she has never stopped interceding on behalf of the world, her intercession obtaining endless miraculous cures and other favors. The role of Mary in the area of obtaining physical miraculous cures is well known and documented in Lourdes, France, this being only one of the many holy places where the grace of God has been manifested through the Virgin Mary at apparition sites.