Zeb is a male name of Middle-Eastern origin.
It is not common in English speaking countries. It was once used in the US around the end of the 19th century. It may be either a name on it's own or a nickname for Zebediah, Zebedee or Zebulun. Zebulun was one of Jacob's twelve sons in the Old Testament.
According to a baby name website, Hebrew translation for Zeb is "to exalt" or "lofty house". I tried using google's translator for both Zeb and Zebulon and got no result.
Apparently, in Arabic the word Zeb is slang for dick. Two years ago there was a Pakistani diplomat who was denied the office of ambassador to Saudi Arabia because his name was Akbar Zeb.
No, I am not making this up.