Here are a few items that I always take on our cruises (as we've learned from previous cruises when we didn't have them with us):
antibiotics (Z-pack) - just in case you come down with something - you really don't want to go to the ships doctor!
ace bandage - in case you sprain something, or worse break a foot - and the infirmary will not see you because you did it on the beach and not on this ship!
benedryl - in case you get stung by a bee on an island or eat shrimp and have a reaction to it (now I carry an Epi Pen).
I bring a few big bottles of water on board. that will last me till the next port.
travel mugs - they're great for your morning coffee and for lemonade around the pool (especially if you're adding a bit of vodka to it).
$$$ - you may not need $$$ on board except for casino and tips, but you do need it.
Electrical tape - for any doors that may bang during the night if the seas are rough.
Paper to write notes to each other when trying to meet up w/other travelers. Tape the note to their door.
something to hang on the outside of your door. it's great to be able to walk down that LONG hallway and easily identify which one is your. We've done martini glasses, parrots, and most recently a christmas wreath because it was a christmas cruise. we also brough one of those collapsable trees and decorated our cabin with it. The gifts that I wrapped for under the tree were small boxes with pictures of the real items left at home. They were light weight to carry and were appreciated on Christmas morning. Lots of fun. Also decorated the cabin w/garland and lights. THe cabin stewarts loved it.
AND don't for your patience and your manners for when the patiences have reached their limits! Everyone is on vacation, but not everyone thinks you should be on vacation. Smile and say, It's a wonderful day, enjoy yours! Then either walk away of steer them the other way.
Most of the stuff yu cn buy, but much of it is at a much higher price (if you can get it from the infirmiry). We walked all over Miami to get to a Walgreen's for the ace bandage. Now it travels w/us.