I don't know if they'd be allowed, but if they are, they might be handy. We didn't realize how much we relied on cell phones til we were out in the ocean with no service. We had to plan where to be and when and had the kids check in at regular intervals. The kids' programs are a good place for your kid to be. We took a cruise on one of Royal Carribean's ships. Apparently it wasn't the biggest, so your mileage may vary, but we were able to run into the girls while they did scavenger hunts and other teen things. You can talk to the staffers to see what they plan to do with the kids, if they'll leave the area, where you could pick them up for dinner, etc. We took a trip where there was only one other toddler that we noticed, so we spent a lot of time with her. We did try their activity programs, but since she was alone, it was kind of a wash. We had a play date with the other toddler but our kid is really gregarious and theirs was shy, so it was only one night. We used the game room.
As far as what to bring, we brought our own sheets for our toddler for the ship's pack n play. I don't know what type of room you have but ours were really narrow and small. We had the pack n play shoved against the wall and the twin beds shoved together for a full bed for me and my husband. The older kids had another room, but they had to be right next to us, per ship rules. We also brought our towels and washcloths for her. We also packed her carseat, figuring that if we took a taxi we might want it. We didn't use it, but I'm not unhappy we had it. An umbrella stroller might be good, too. Ships are narrow and little kids get tired. Also, if your 3 yr old is still potty training, find out if he/she might not be allowed in the pools. Our ship did not allow non-potty trained kids in the pool so that really changed some of our plans for the family. We couldn't all swim at the same time. I did take swim gear, though, as I knew we'd hit the beach when we got to Bermuda. We also brought a few snacks for the little one because I didn't know what she'd eat at dinner. We had the option of going to the formal dining room for all meals, but ended up eating breakfast and lunch cafeteria style so we didn't have to wait. Dinner was our only sit-down and order meal. The staff also got used to us and started having fruit ready for our daughter. If you know the toddler might melt, ask for something off the menu for him/her to get started.
Sunblock. It's deceptive to be on a ship but you'll be in the sun. We all got burned from not thinking about being in/out of the sun all day.
EVERYTHING on a cruise is billed to your room, even if you want to pay cash. It's sneaky so you don't realize you're paying so much. You'll get a bill at the end of the trip. There's also tips for everybody, usually a standard rate for all like $3 per day for your waiter at dinner. Factor that in your budget. We told the kids that they had to tell us before they charged anything extra or they'd be responsible at the end of the trip. They were pretty good about it. In our case, room service was free, but ice cream at Ben and Jerry's was not and so long as they asked, we were ok. We bought the soda plan where for a flat rate you could get soda at any bar and at meals.
Bring a couple of nice outfits. We had a dress code at dinner and one night was formal. Also, they may take pictures so be prepared for that. It was really nice to have a formal picture of the whole family.
If anyone gets motion sickness, ask your doctor for dramamine patches. They go behind the ear and stick there for a couple of days. My stepson used them and they really helped.
If you do excursions, consider doing one or two planned ones and a couple of on your own things when you get there. We missed the glass boat snorkel trip, asked the locals, and ended up at a beach where the fish were terrific, and we could all get out and swim vs the toddler and I stuck on the boat while others swam. We got bus passes and wandered around on the island for two days.
If you drink alcohol, find out the ship rules. Some may not allow you to bring it back on board.
If you buy flowers, know that they will probably be confiscated by AFIS on your return. When we left the boat, all the pretty flowers were on the side of the exit because they weren't allowed back in the US.