Why is the name of the mother preferred in blessing the sick child? Is it not a patricial system in Judaism as we recall Abraham as the first father of the nation?

Judaism is not a patriarchal system, but rather there is importance of both the father and of the mother, e.g. even one's Judaism is decided by their mother! That's why, as opposed to all other ancient religions, Judaism often mentions the 4 fore-mothers (Sara, Rebecca, Rachel and Leah), together with the 3 fore-fathers. In prayer for the sick we wish to particularly evoke mercy and emotion, by us and as if it were, by God, as well, and these are considered more feminine traits, so the way of the world is to turn to the more feminine aspect (e.g. Tehilim 116, 16, "Please, O Lord, for I am Your servant; I am Your servant the son of Your maidservant..."). Similarly, our rabbis teach that the prayers of women are especially effective. Accordingly, when one is God forbid sick, we pray using the mother's name.