I bring olives home in jars. We travel to mainland Europe about 6-8 times a year and buy olives as we need them. From Provence, from Tuscany and Liguria, from Spain and Portugal. I don't keep any of them for long!
The rice? Well, for Greek dishes, the rice has a dry and nutty consistency. I like it.
I forgot to say, when we go to Corfu or Cyprus, we buy lots of local honeys, too.
Every region of greece is known for something. Crete is known for olive oil, Syros for its turkish delight, Egina for its pistacchios, Lefkas for its lentils, Santorini for its yellow split peas and wine, etc. Ask your family to find out what Zakynthos is known for and ask for that. I can tell you though that the islands in the region where Zakynthos is not at all well known for its olive oil so you're taking your chances there.
My advice is to visit the tourist shops but don't buy food items there. There you will find abundant examples of what flavors the island is known for. It's research. Then go to the hotel concierge or find local people that can tell you where THEY buy things.
For things like honey, vinegar, and olives I've never had a problem bringing them back as long as they were well wrapped.