Favoriting tweets tends to be a bit idiosyncratic- everyone uses the feature differently. Some use it like Facebook's "like" button, some use it to save links for later, and some use it in far craftier ways.
If you actually want to do something with those tweets, you have some options: Pinboard and Gimme Bar, for example, can both pull in your favorited tweets so you can browse them later. If you want to further organize them from those services, you can use Pinboard's tagging or Gimme Bar's "collections" feature.
What if you wanted something simpler? That's where Tweetbits. Tweetbits is a simple service that collects your favorited tweets, then lets you come back to them later and organize them into "bins" with an easy drag-and-drop interface.
Currently, you can only view the list of your favorite tweets, but the "Your Twitter Feed," "Images," and "Export" tabs (which currently do nothing) hint at things to come.
Granted, it's not the prettiest of interfaces, but the service certainly keeps it simple while giving you an easy way to organize the best of your timeline. Because of its simplicity (and its speed), the service is a bit more versatile than other mechanisms. You could keep a list of, say, inspirational quotes in one bin, articles to read later in another, and jokes you want to use at that dinner party later in another. Tweetbits is currently in beta, so if you have suggestions for improvement, they're listening.
There are plenty of services that integrate your favorited tweets into their catch-all buckets, but I've yet to come across anything as simple or as easy to use as Tweetbits.