I'm sure I'm not the first person to suggest this, but here goes anyway:
Twitter is ubiquitous these days. News, tech, sports, celebrities, music events - Twitter and hashtags (etc.) are where EVERYONE finds out what's happening and what other people think about what's happening. Quickly and succinctly (140 chars etc) in real time.
From TV stations to radio to Internet hubs to the man in the street, Twitter is THE place to go - and yet we keep hearing that Twitter is struggling to make ends meet in terms of making a profit and so we have to put up with ads and sponsored tweets.
WHY DOES EVERYTHING IN THIS WORLD HAVE TO BE ABOUT MAKING A PROFIT? When something's genuinely and universally useful, why couldn't someone/something rich (think Apple, Google, or even the USA government) step in and fund Twitter out of their small change - the amounts concerned are relatively tiny.
Say 500 million dollars a year to staff and run. It sounds a lot to you or I but it's nothing to the big boys. In fact, I'm amazed that Google in particular hasn't bought up Twitter by now, it's such a natural fit.
Or - thinking laterally - I think Twitter could get 'there' using public support. 350 million users and I know a lot of the typical members of the public want everything to be free, but I'd bet that there are a LOT of users (including me) who would happily pay (say) $100 a year to keep Twitter going.
So, doing the maths, if only 1 in 70 Twitter users chipped in to this level, on average, then Twitter would be completely funded.
Just thinking out loud. Twitter is finding it hard to 'monetise' because its core service isn't about products, it's about real time one-to-many communications. My point is that it shouldn't HAVE to monetise. At all.