soccer:

1889, socca, later socker (1891), soccer (1895), originally university slang, from a shortened form of Assoc., abbreviation of association in Football Association (as opposed to Rugby football); cf. rugger, but they hardly could have taken the first three letters of Assoc.

football:

the open-air game, first recorded 1409; forbidden in a Scottish statute of 1424. The first reference to the ball itself is 1486. Figurative sense of "something idly kicked around" is first recorded 1532. Ball-kicking games date back to the Roman legions, at least, but the sport seems to have risen to a national obsession in England, c.1630. Rules first regularized at Cambridge, 1848; soccer (q.v.) split off in 1863. The U.S. style (known to some in England as "stop-start rugby with padding") evolved gradually 19c.; the first true collegiate game is considered to have been played Nov. 6, 1869, between Princeton and Rutgers, at Rutgers, but the rules there were more like soccer. A rematch at Princeton Nov. 13, with the home team's rules, was true U.S. football. The earliest recorded application of the word football to this is from 1881.

rugby:

1864, after Rugby, public school where the game was played, from city of Rugby in Warwickshire, central England. The place name is Rocheberie (1086) "fortified place of a man called *Hroca," second element from O.E. burh (dat. byrig), replaced by 13c. with O.N. -by "village." First element perhaps rather O.E. hroc "rook." Rugby Union formed 1871. Slang rugger for "rugby player" is from 1893.