Almost 50% of the children have brothers or sisters. So many people know what it means to have a brother – to have someone you might hate sometimes, because he probably competes with you sometimes, but if you need him, he will be there for you just because he is your brother. Especially if he is your big brother: He is more grown up and stronger than you and your rivals and is able to protect you well. So why the hell do you call a dictator, a leader of a totalitarian system “Big Brother”?
Well, in “1984” George Orwell uses a lot of antithesises like “War is peace” etc. to underline the twisting of the reality into the reality the government wants and this surpressing, controlling and brainwashing leader is also basically the antithesis of his name – The way he acts is in contrary to the way he presents himself to the citizens and how everyone sees him: As a protector. Someone who is caring about them, who vaporizes the evil ones. And since the leading of the totalitarian system is represented by “Big Brother”, by one subject, the people can put trust in this person. But probably this face that is presented to the people is not even the face of a real person, its just the fictive incarnation of the collective domination of the inner party.
So Orwell chooses this name to give a frightening proof how successfull totalitarian systems can present themselves as the helping hand, the good guys or the big brother taking care of the people.