I'm afraid you've probably got the right dictionaries, but the wrong name book (on this occasion!). Fabis is wildly imaginative ... if it means anything, it means: With Beans! ... you couldn't get Fabius from it anyway, not without major surgery. Fabius was a name, there was a great general called Fabius and known as FabiusMaximus, and the Fabian version (Fabianus) is an adjective form of his name - it would have been used by his slaves, if he freed them, or possibly by people, soldiers who had served with him perhaps, who wanted to commemorate him - but there are lots of gaps in our knowledge of how the Roman naming system worked - especially for girls.
I suspect the author of your book was either ignorant and lazy (harsh words, but often well deserved) or honestly misled by some ignorant, idle predecessor. Or was too sentimental to consider tasty, honest, nourishing beans to be worthy items to be named after!