I remember the Alliance Tournament initially being sponsored by the Caldari Gaming Commission.
A lot of the sports in Eve have struck me as probably being more technology oriented such as with Mindclash or Splinterz. This might have to do with that with all the potential human biomedical augmentation, cybernetics and pharmaceuticals available it's probably hard to ensure a truly even playing field -- steroids and other performance enhancing drugs are already an issue in today's sports, who knows how difficult it might be to prevent similar abuse given Eve technology. This might account for the popularity of Mindclash and Splinterz, they offer or at least offer the perception that the competitors are all on an even playing field.
Beyond that there might be a lot of digital sports that are popular and have a wide fanbase. I'm sure there's New Eden equivalents of competitive LoL/Dota/SC2/FPS games that would draw people in.