Autoupdate/refresh of tweets (every x minutes) would definitely be on top of my list. Birdhouse integration or similar in-app functionality a close second.
Multiple ways to do Notifications, including Push, email, perhaps SMS, and, hopefully sooner rather than later, the ability to run it in the background.
I would enjoy a more robust lists feature, with list/search list creation on the fly that's more intuitive/less options during creation, but more custom options in settings.
I've tried the lite app and I like the fast velocity of animation and action.
I'd like to see the possibility to post currently playing music, major integration with drafts message and integration with birdhouse