Why do they call him “Bibi”?
It’s an affectionate nickname for Binyamin.
A good nickname is hard to find. Most politicos would sell their grandmother for a winning tag like “Bibi.” Netanyahu didn’t even have to pay a public relations pro to think it up. According to Netanyahu family lore, it stems from childhood confusion with another older Benjamin who was the big BB while the future prime minister was the little Bibi.
http://www.jweekly.com/article/full/3680/netanyahu-nixes-bibi-but-headlines-scream-for-it/