The Origin
Bananaphone started as a song in the 1994 children's album Bananaphone by Raffi, a children's singer.
Original Song:

Although the song was released in 1994, the Bananaphone Meme wouldn't peak until 2004.

A Meme is Born

The folks at Cookiethievery.com increased the pitch of the song by six semitones and created The Osakaphone flash for it, featuring a Gundam and a JIS art rendition of the character Osaka from the anime Azumanga Daioh.

The first known appearance of Osakaphone was on Applegeeks.com on May 1st, 2004.

On May 4th, 2004 a Flash cartoon entitled "Bananaphone" featuring the Raffi song was created and uploaded to Newgrounds by users Dave Teatro and LazyWill.

Bananaphone

The next day, May 5th, the Osakaphone flash was posted to the site Transbuddha.com

Variations on both flashes quickly emerged, and it was no longer clear who was imitating who.

Different versions of the flash meme were popular on NewGrounds, SomethingAwful, 4Chan, Ebaum's World, Albinoblacksheep, and more.

As video-sharing sites like Youtube became more popular in 2005, many other videos and flash animations where created.

Harry Potter Version:

Star Wars Version:

Badger Badger Version:

However, we can see that the idea of a phone that is also a banana was also used in other places, like Sesame Street:

For more information on Bananas, see Peanut Butter Jelly Time