My etymological name book says that Emerentiana (this name occurs once in my family, btw) comes from Emerentius, which in turn indeed derives from Latin 'emereo(r)', which means 'I wholly deserve' (according to my name book).
The book says that the holy Emerentiana was a
Roman martyr, probably during the prosecutions of emperor Diocletianus. The day of Saint Emerentiana is January 23rd.