I found the meaning of "Fabian", but not "Fabius". Yes, Fabian does have something to do with beans:
FABIAN Gender: Male Meaning: Bean Grower Origin: Italian
* By the way - there might be a chance that there is some Greek origin somewhere, because there is a type of Greek food called "Fava" in Greek, but probably spelt "Faba" in English or if translated into English. Thsi is a type of bean mash.