PropDad:
There are more than 12 emoticons but we are trying to limit it to 12 only. Even if the button just cycles to the next one. Is this feasible the the Arduino? I am trying to find out before I purchase anything.
This is quite feasible. However, you aren't limited to that.
The main limitation on how complex the input can be will be the total I/O pins, as you'll need some pins reserved as outputs for your LED matrix. With one or more buttons you could do as you describe; have some sort binary code where pressing one to four buttons simultaneously picks a specific emoticon; or, depending on which Arduino board you are using, you might have enough pins for each emoticon to have it's own button. With a different input device, like a rotary encoder, you could use much more than tweleve emoticons with just a couple of input pins. The only potential drawback is that even a relatively inexpensive rotary encoder will probably cost a bit more than twelve common push-buttons.
There really are a multitude of different input options available, check out the mechanical inputs section of the Playground for some ideas.