Android Jelly Bean introduced a "slide to hide e-mail notification" feature, an improvement on the "click the little X to hide an e-mail notification".
This new feature still has a lot of richness that can be packed into the same space and gesturing.
For starters, it actually leads the user to want to have 2 possible end-game scenarios for any particular notification: slide-left for X effect, slide-right right Y effect. At the end of either X or Y effect, the notification can be optionally resolved and cleared away from the notification bar, if applicable.
Further observation will note that a default "slide all right" button exists at the top of the notification pull-down screen (aka notification center -- what is this screen's official name, anyway?). This behaviour should always simply be the same old "dismiss notification".
Initially, I wanted to simply propose that Gmail notifications could get a "slide left to archive" behaviour -- as a "+1" enhancement -- but this is probably too specific... so let's propose a more general purpose enhancement!
To push this concept further, it would be wonderful if "slide-left" actions could be customizable for Gmail, and to push even further... be customizable for all apps. For example, "slide left to reply".
Then, maybe as "+3 or +4" level of enhancement, it could become "slide left, notification presents [archive], [reply], [forward], [mark as spam]; tap action icon (or words); action is launched, meanwhile notification is dismissed".
Another advanced scenario that could work well is in music player apps. For example: notification center is used to display "currently playing" info (ie: title, artist). Slide-left pulls in a collection of controls - play/pause, next, previous. The notification does not need to be dismissed, it can be programmed to stay in "slide-left sub-menu mode" for X seconds before reverting the standard notification mode, or slid back by doing a slide-right.
Was the text description enough? Or would visual aids be needed...?