She's rich in her own right.
Without seeing the source of this quotation in context, the closest way to describe it, would be 'She is rich' regardless of some other qualifier.
When quotes are taken out of context it is difficult to give an exact explanation.
Imagine a twenty-year-old female entrepreneur who has a successful clothing brand, and she also has wealthy parents.
To say 'She is rich in her own right' means she has her own income, and she does not rely on her parents for financial support.