2Vermont,
Anyone can baptize. You can validly baptize your mother as long you intend to do what Christ wants done and follow the instructions. Use the proper form (words) and matter (natural water). The same person that is pouring the water must say the words. Remember that it is in your power to open the gates of Heaven for her. Do not take this slightly! If you have to wait until she is dying and unconscious, so be it!
Of course, the first thing you should do is call a priest to baptize her when she is unconscious. Given that she is not a Catholic, and if a priest is not at hand, the process to follow is to recite slowly in her ear the Act of Faith, and the Act of Perfect Contrition, and then baptize her.
1. Act of Faith
O MY GOD, I firmly believe that Thou art one God in Three Divine Persons, Father, Son and Holy Ghost. I believe that Thy Divine Son became Man, and died for our sins, and that He will come to judge the living and the dead. I believe these and all the truths which the Holy Catholic Church teaches, because Thou hast revealed them, Who canst neither deceive nor be deceived.
2. Act of Perfect Contrition
O MY GOD, I am heartily sorry for having offended Thee, and I detest all my sins because I dread the loss of Heaven and the pains of Hell; but most of all because they offend Thee, my God, Who art all-good and deserving of all my love. I firmly resolve, with the help of Thy grace, to confess my sins, to do penance, and to amend my life. Amen.
3. Baptism words while pouring the water on her forehead:
"I baptize thee in the name of the Father, and of the Son, and of the Holy Ghost".
You could also say it in Latin:
" Si non es baptizatus (-a), ego te baptizo in nomine Patris, et Filii, et Spiritus Sancti"
Do not cease in praying for the conversion of your mother so she willingly accepts baptism and becomes a member of the Mystical Body of Christ.