You maybe tempted to think this phrase is about making the isla (island) of Cuba a better place. After all, Cuba needs no translation, and libre means free. While that is a great idea, that's not what our phrase "cuba libre", is refering to.
I picked this up in the Dominican Republic after having my second or third "ron y coca" (rum and coke). The bartender was nice enough to tell me that I only need to ask for a "cuba libre", and that everybody knows what this is.
It turns out the bartender was right. I've been to Mexico several times and ordered many a "cuba libre", served with a "limón". Something else I discovered (the hard way), is that the usual translation of "limón" is lemon, but if you ask for a "limón" in Mexico, you'll get a lime. You'll have to ask for a "lima" or a "limón amarillo" if you want a lemon.
If your drink (trago) of choice is a "cuba libre", then your Spanish will sound just a little more authentic when you order your next trago.
¡Hasta la próxima!