Why are Britain's elite private schools called public schools?

These schools were (and are) public in the sense of being open to all students in principle, though at the time of their foundation most older schools were run by the established Church and were only open to boys of the same denomination. In practice however many such schools are highly academically selective and pupils usually need to pass the Common Entrance Examination before being admitted at all, and all but the best scholars must be able to afford the considerable fees for tuition and (for boarders) room and board.

https://en.wikipedia.org/wiki/Public_school

The term "public" (first adopted by Eton College) refers to the fact that the school is open to the paying public, as opposed to a religious school, which was open only to members of a certain church. It also distinguished it from a private education at home (usually only practical for the very wealthy who could afford tutors)

https://en.wikipedia.org/wiki/Public_school_(United_Kingdom)

A public school is an independent secondary school which is a charity (not profit-making) and which belongs to one of the public school associations, the largest of which are the Headmasters' Conference (HMC) and the Girls' School Association (GSA). The expression "public school" can be confusing: in many countries other than England a "public school" is a school which is run by the government, which is not the case with these schools. In England the term private school is used to refer to any school which is run to make a profit. Among the most famous public schools are Eton, Harrow and Winchester.

http://www.ukstudentlife.com/Course/Boarding.htm

As has been said, originallythey were schools for the public - with public having a rather limited definition. (A bit like the difference between society and Society...)

http://english.stackexchange.com/questions/8144/difference-between-society-and-the-society

In the UK public schools are private schools, but not necessarily vice versa. State schools are usually run by local authorities not the government, and most church schools are part of the local authority educational system for all practical purposes. City academies are run by private bodies but paid for by the government - and count as local schools but the local authority has nothing to do with their administration. Confused? You try working in the system. Source: attendance at a minor public school (founded for the education of local boys not too far off 400 years ago) and over a decade of working in the local authority educational system.