I've come across the female name
Alivia used twice in medieval Britain. One was
Alivia de Hamme/de Hame (that was her married name anyway), and the other was
Alivia Braose. Both lived in the 14th century.
Alivia Braose was also called
Alice. Was
Alivia a 14th century variant of
Alice? I doubt it was related to the Shakespearean name
Olivia.