I tried the first part of [this tutorial with a LilyPad Buzzer and a LilyMini. It caused the LilyMini to freeze. I had to bring it back to life using [recovery mode.
And now the port I used (number 1) doesn’t do anything anymore. Did I fry the port simply by trying to use the tone() function?](Programming the LilyMini - SparkFun Learn)](LilyPad Buzzer Hookup Guide - SparkFun Learn)
Apparently port 1 can only be used as an analog port. It doesn’t support digitalWrite. 
But also using other ports, the tone() function causes the LilyMini to freeze up.
I have also been wondering whether the LilyMini can support the buzzer. My thinking is the biggest issue would be what Voltage the buzzer uses. Since the LilyMini only uses a single 1.5V battery, it might not work with the buzzer.
The voltage isn’t the problem. The LilyMini outputs 3 volts. I was able to make a sound by switching the buzzer to HIGH and LOW rapidly. I guess I could write my own tone() function.