Twitter Co-Founder Biz Stone has been quoted as saying that the bird logo is named after Indiana Legend, Larry Bird!
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Acoording to, Stone grew up in Massachusetts during the time that Bird was playing basketball for the Boston Celtics.  We can probably assume that he was a big fan and that is why the Twitter bird's name is Larry Bird. 

Larry Bird grew up in French Lick, IN and then went on to play college basketball for Indiana Sate University before being drafted by the Boston Celtics. Fun fact, Larry Bird was a physical education major at ISU. He did his student teaching at West Vigo High School, my alma mater. I was only in middle school at the time, but we went to all of his baseball games. Yes, I said baseball games. He student taught during the Spring semester. Basketball season is in the Winter. He helped couch football while he was student teaching, It was so cool to see him on the baseball field. Remember, this happen right after he took ISU to the final game in the NCAA Tourney, only to lose to Michigan State and his long time friend, Magic Johnson.