I think your best option is a Guayabita de Pinar. It's a spirit made with tiny guava fruits, it's nice and sweet. I'd choose the one made by La Occidental. Here's a good picture of it from flikr.
Otherwise Aguardiente de Cuba is an option. Aguadiente means firewater, and it's not nearly as friendly as the Guayabita. My understanding is that it's related to rum in the way that mescal is to tequila. Funky. I'd only go for it if you have a decent amount of cocktail experience and you'd like to experiment. Ron Santero makes the most popular bottle.
Aside from that, you might consider something like a local soda, or some Coca-Cola. Coke changes the recipe depending on where it's shipped, so if you want to drink a proper Cuba Libre ask for some.