Mystery solved as to why they called it ELEPHANT.

http://en.wikipedia.org/wiki/Elephant_(film)

Title

The title is a tribute to the 1989 BBC short film of the same name, directed by Alan Clarke. Van Sant originally believed Clarke's title referred to the story of several blind men trying to describe an elephant and each one drawing different conclusions based on which body part they were touching. Later, he found out that it was referring to the phrase "elephant in the room", a reference to the collective denial of some very obvious problem. Van Sant's film uses the earlier interpretation, as the same general timeline is shown multiple times from multiple viewpoints. Also, a more subtle allusion is to the elephant as the symbol of the Republican Party, which becomes relevant insofar as the film raises political questions about gun laws and homophobia in America.

The earlier film reflects on sectarian violence in Northern Ireland. Van Sant's minimalist style and use of tracking shots mirrors Clarke's film.[2][3]

A drawing of an elephant as well as an image of an elephant on a throw on the bed can be seen in Alex's room, while he plays the piano.