Pinned tweets
If you use Twitter, you’ve seen promoted posts. They are essentially a form of random spamming on Twitter’s part, routinely irrelevant and hardly ever interesting - nothing more than an annoyance.

On the other hand, there are some tweets that I actually want to see. For example, there are news and updates posted to company’s accounts. These are infrequent, released unpredictably and they are exactly the reason why I am following these accounts in the first place. But they get lost in the noise.
What Twitter needs is an ability to promote tweets to the followers.
The idea is to reserve the top part of my tweet stream for tweets that are promoted, or pinned, by their posters. This way if I go on a vacation and a vendor ships an update, I will still see the news when I get back:

Once a pinned tweet is shown to me, it starts dropping down in my timeline as usual. Think of it as a delayed, on-demand tweet delivery.
It also makes sense to allow posters to un-pin their tweets to allow delivering short-lived, expiring news such as this:

And that’s it - an intuitive and useful extension to the Twitter mechanics.
Also consider this:
Pinning essentially lets posters mess with their followers’ streams, which is something that requires a great degree of restraint and diligence. As such it has a property of self-correcting the abuse. You start pinning too many tweets - you will see me promptly unfollow you.
Secondly, pinning is a form of relevant advertising that is actually useful. I am an ardent AdBlock user of several years, and this is the first time I am actually asking for ads. Take a second, let this sink in :)
Lastly, this should be very easy to monetize and on an absolutely massive scale. There’s not a single person who wouldn’t want or need to pin a tweet now and then, especially if it’s an affordable thing to do.
† Yes, there are Twitter lists, and, No, they are just not convenient enough.
‡ A discussion thread - http://news.ycombinator.com/item?id=3971933
Update
The “#! tweets” is the Version 2 of the same idea. Have a look.