I know that, on the game summary screen, I can see when my first win of the day bonus is available. Is there a way to see that countdown that doesn't involve playing an entire game?
With the Hecarim patch, this functionality has been added:
Added first win of the day status to the summoner profile.
To access it, simply click the tab to view your profile. On the main screen where you can see your various rankings, the main banner will indicate the status of the daily bonus on the lower right.