Wikipedia gives this:
Leyton Orient was originally formed by members of the Glyn Cricket Club in 1881, many of whom were former students of Homerton College. The team has had several name changes since, first as Eagle Cricket Club in 1886 then as Orient Football Club in 1888.
The O's are the second-oldest league club in London behind Fulham, and the oldest London club playing in the Football League following Fulham's promotion to the Premier League. They played in the 2nd Division of the Southern Federation's League in 1904, joined the Football League in 1905 and are the 24th oldest club currently playing in the Football League.
History books written on the club by its historian Neilson N Kaufman suggest that the choice of the name Orient came about at the behest of a player (Jack R Dearing) who was an employee of the Orient Shipping Company (later to be taken over by P&O). The club's name was changed again to Clapton Orient in 1898 to represent the area of London in which they played, though there was another team called Clapton F.C.
The name Leyton Orient was adopted following the conclusion of the Second World War. The club had moved to Leyton in 1937, though again there was another team called Leyton F.C. A further rename back to simply Orient took place in 1966 after the Borough of Leyton (in Essex) was absorbed into the London Borough of Waltham Forest. That renaming followed a financial crisis (one of several to hit the club, and by no means the first or last) and restructuring of the company behind the club; this is remembered for a "pass the bucket" collection that took place at a special meeting of supporters in the East Stand, when complete closure was claimed to be a definite possibility.
The club finally reverted to "Leyton Orient" in 1987, shortly after Tony Wood took over as chairman and at a time when a supporters' campaign was taking place in the Leyton Orientear fanzine to reinstate the Leyton part of the club's name.
