Bring more kids clothes than you need. Things will get wet, things will get messy, and the only self-serve laundry you have will be a tiny bathroom sink. Consider packing outfits in large ziplock bags: you can put wet or messy clothes in them to take home without ruining your luggage.
Do not assume that the sandals that fit your child in August will fit them in January.
Pack too many diapers. Make peace with the fact that you have packed too many diapers. The ship will require them if you want to take him in the baby splash zone. The towel station probably has some, but it might be easier to know in advance what size you need.
Bring diaper cream, bring children's tylenol, bring plenty of sunscreen.
Prepare for snacking! You can absolutely take bowls and plates from the buffet into your kitchen, but you'll want ziplock bags to sneak cheerios onto the beach. (No, it's not allowed. Everyone does it anyway.) Bring an extra sippy cup or two. If you'll be giving the kiddo milk, bring a tiny bottle of dish soap to wash your cups out.
The buffet is easier in some ways, but it's actually nice to have full-service dinners. (Unless you enjoy getting up sixteen times every meal to fetch things.) If you're interested, and you don't want to eat late, reserve your dining time ASAP.
They ceilings are magnetic - you can get magnetic hooks on Amazon for surprisingly cheap. On the advice of someone on the boards, I brought a shower curtain and used it to rig up a divider between our bed and the kids' bed. Ours seem to have an easier time falling asleep when they can't see Mom and Dad do exciting things like reading books or hanging up swimsuits. Magnetic hooks are also great for hanging up swimsuits and such from the ceiling.
Space will be in short supply, especially if your kiddo is big enough to need the sofa bed. Consider bringing your smallest stroller, your smallest car seat. (You don't need one of the expensive fits-on-a-plane strollers, but don't bring the jogger that takes up as much space as an armchair while folded.) Personally, I wouldn't bring sheets or blankets unless your child is especially precious about them.
The kids club will let you check out a small bag of toys to take back to your room. However, they will charge you if you lose pieces. My two year old dropped at least one car off our balcony on our most recent cruise. Because it was a $0.50 hot wheels car and not a $50-to-replace cruise-owned component, we didn't have to drop everything and rush down eight floors to retrieve it from the deck.
Do you drink wine? Royal will let you bring two bottles on board. A glass of wine on the balcony during nap is really nice. You can also carry drinks down from the bars to your room, but it's more expensive and less relaxing.
If you have some sort of hearing protection for the child, you might want to bring it. The shows are usually family-friendly, but volume tends to cater to the "I saw the Beatles live when I was your age" crowd. You might get to enjoy something together, instead of leaving with a screaming child or being stuck in your room in the evening.
There's plenty of stuff you could bring. A shovel and collapsible bucket for the beach? Inflatable bolsters to keep your kid from rolling out of bed? Fun magnets, so you can pick your door out of the rows of identical cabin doors? Honestly, you don't need most of it. Be patient, and accept that you won't be able to do as much as you would have pre-kids. And bring extra diapers.