The English word is borrowed from French julep; the medieval Latin is julapium. This Latin word is from the Arabic julāb, which is in turn from the Persian gul-āb rose-water. You can see the root "gul" in the title of a famous work of medieval Persian literature, Gulistan - "The Rose Garden" - by Sa'di.
Below is an image of a mint julep in a silver cup:
Now, truth be told, while I am not a big fan of mint juleps, I am a very big fan of the Indian sweet called gulab jamun (you can see gulab/golab word in there) which has a rosewater syrup as one of its ingredients. Yum!