e.g.
Yacov Menashe Ben Rachamim
Yehudit Bat Noah
David Ben Ze'ev Ariyeh
Gam Bashan veGam Gilad
Thanks
For start Gam Bashan veGam Gilad is a forum nickname meaning Both the Bashan and the Gilad, (Two Israeli teritories east of the Jordan river that are currently occupied by Syria and Jordan).
and the answer to your question is that many Jews have two personal names. For example many Haredi Jews have a hebrew name and a Yidish name, many other Israelis have two hebrew names, the second of them may be that of a dead relative. Jews in the exile often have an English name and a Hebrew one.
Usually only the first name is used however their full name in formal name is for example Meir David Kahane. Ben (and also Bar) means so or son of, so many hebrew family name are of the form Bar David, Ben Harush, ... Bat is the feminine form of ben however as a family name we use Ben/Bar both for women as well like: "Datia Be-Dor"
Moreover Ben is also a personal name in modern Israel so we have Israelis with names like "Ben Cohen" and Ben is their name not a shortening of Benjamin.
Finally names Like Chaim Ben Pesach as are common in our forum are compounds of:
{the personal name(S) of the person} + Ben (son of) + {The personal name(s) of his father}, this is the traditional form of Jewish naming since before use of surnames has begun and this is how we still use in prayers, I hope it answers your question.