The Bocchicchios were a Mafia clan from southern Sicily, who moved over to America and became mediators for the other families, after many of them were culled or exiled by Benito Mussolini. They settled in the Hudson Valley, near New York city.
While not the most intelligent immigrants from Sicily, the Bocchicchios did have very strong family bonds, and as such they took the Sicilian concept of vendetta to an extreme, considered fanatical even by Mafia standards. If one of their own were to be wronged, the whole Bocchicchio family would stop at nothing to take revenge upon the one responsible, and they never forgot a wrong. The offender would find himself constantly under attack by an near-unending wave of vengeful Bocchicchios who can't be bought off, are undettered by threats or armed bodyguards, who would search to the ends of the earth and even commit suicidal tactics nonstop until the perpetrator was dead.
It is this kind of vengeful nature that has actually been their greatest source of income, renting out a hostage service to more powerful families.
If a negotiation or sit-down was going to take place between two or more families, then, as a way to show complete sincerity, the Boss who called the meeting would pay the Bocchicchios a large fee for one or more hostages. This exchange was a promise by the client that the hostage would come home alive. The hostage would then be put into the custody of the other families participating until the meeting was over and the family's negotiator returned safely, at which point the Bocchicchio hostage would be let go. If the negotaition turned violent and the negotiator was killed, then the hostage would be killed, and the Bocchicchios would seek revenge, not on the hostage-takers, but on the family who had killed the negotiator, reasoning that that family was responsible for the hostage's death.
One example is when Michael goes to meet with Sollozzo and McCluskey. Sollozzo hires a "negotiator" and sends him to Clemenza's people to ensure Michael's safety. (Clemenza comments his men are letting him win at pinochle.) Should something have gone wrong and Michael be killed, Clemenza would then order the death of the negotiator. The Bocchicchios would then seek revenge on Sollozzo, rather than Clemenza, because it would have been Sollozzo who had broken the trust.
In the Godfather's Revenge
The Bocchicchio family could be based on the Morello family, which also consisted of mainly family members and was once a powerful organization. They could also be loosely based on DeCavalcante crime family, who acted as mediators during the Bonanno family war in the 1960s.