There are multiple origins and meanings for the name:
1. Derived from the ancient Greek words μόνος [monos] "only; only one" + νίκη [niki] "victory" = "lonely victress"
2. See Monika/Manika
Saint Monica, the mother of Saint Augustine. In fact everything we know about Monica comes from her son and most of it in the context of his own biography. The odyssey of their lives was closely intertwined for over thirty years. Some have seen her intervention in Augustine's life and her steadfastness as typical of a domineering mother, but Augustine in retrospect chronicles her heroic struggle to bring him not to herself but to Christ.
Her feast is celebrated on 27 August, the day before that of her son, Augustine.