There is no shortage of souvenirs to remind you of your trip to Cyprus. The local pottery is lovely (visit the pottery in Lemba for the best we've found yet). Leather goods are another possible souvenir or you might like to take a trip to Lefkara to see the famous lace being made. You can buy it in the shops around Paphos Harbour too.
Don't forget Cyprus' fine wines. They are well worth trying as are the local after dinner drinks of Commendaria (fortified red wine as drunk by the Knights of St John) and Filfar, a cointreau type liqueur.
For day to day shopping the supermarkets are very good, very reasonably priced and will stock pretty well anything you need or want. You'll find all your familiar brands plus some unfamiliar looking local delicacies.
If you have a sweet tooth, don't forget to take back some Cyprus delight - you can even go and see it made in Yeroskipou just outside Paphos.