We cruised with my son to Bermuda at 16 months, so I know that is a lot older, but this is what I would recommend/learned from our trip:
Take an extra swim diaper or 2: They don't take up much room, you will likely only need 1 per beach trip, but if one gets dirty, you will want an extra. I would take 2 swimsuits for him max: the one from day one would be dry by day 3 in Bermuda.
Diapers: They are EXPENSIVE on the ship. We spent $35 on a package of 30 I think. Overestimate what you will need, then add at least 6 more, haha!
Cabin: Will your room have a dividing curtain? If not, we used super strong magnetic hooks to hang a dark, thick fabric shower curtain around the pack n play. It went around a long side and a short side, so when it was in a corner, he had a cozy little room of his own. We also took a white noise machine (my son sleeps with one so we had it already, an app on your phone is another option).
Stroller: Umbrella stroller, anything bigger is going to be hard in the hallways.
Diaper Bag: I had a big backpack for days off the ship, and a small tote bag for on the ship with just a diaper, wipes, a small toy maybe for going to dinner. You are so close to your room, you won't want to lug a huge bag all over.
Lillebaby: Don't leave home without this. Very helpful in St. George if you go up there.
Inflatable bathtub: We had one and didn't use it. We found out at our Pre-Cruise hotel in Boston that our son likes showers. I hadn't even thought to try that ahead of time. So I would try showering with him at home and see if he likes it, it would be one less thing to haul if he tolerates showers.
We took a baby monitor, but we had family in the cabin next door who were willing to listen once he was asleep if we wanted to go to a show/grab a drink. If you don't have a balcony or other family, you won't have a use for this.
Things you might think about: A couple of plastic things for beach toys, we took a dollar store sand shovel and a plastic cup and he was as happy as a clam.
You know your kid best, but I didn't need new PJ's every night. I would take 1 sweatshirt for cool areas on the ship but not multiple.
We really didn't use toys much at all, he was so busy looking at everything and everyone, I would pack light on that one.
We took gripe water as well, it helps with various tummy issues and has ginger which is soothing. Just something to consider.
We LOVED Bermuda for our son's first cruise, it was so nice to be in the same port and not be in a rush to get back each day. I felt less pressure to get off the ship the moment we could each day because we had so much time to be flexible. And this ship's staff were incredible and nice and helpful! Hopefully you will have a great time!