The 'Ivy League' as an officially recognized component of the NCAA
may, in fact, have been around since 1945
or so. However, the roots of the name travel back much further. Originally (I believe around the turn of the century, although I'm not positive) when universities began to actually play tackle football
as well as other sports in an organized intramural
fashion, some of the oldest and, more importantly, largest at the time formed a league
to regulate their competition.
Harvard, Yale, Princeton and Columbia were the original members. There were four of them, and the League was originally dubbed the League of IV. This was quickly shortened to the much less cumbersome 'Ivy League' especially considering the obvious pun. The use of 'Ivy' to describe the other members of the current fraternity (Brown, Dartmouth, Cornell and UPenn) dates to the formation of the NCAA division, which uses the English variant of the name.