We too are a family of 4 and found that we really needed:
1. power strip - with all of the electronic gadgets we have this was an absolute need to recharge things!
2. over the door shoe organizer (esp. on 7 day - not so much on 4 day) - this helped keep things like sunblock, hairspray, medicines, deodorants, toothpaste & brushes, etc organized and off of the limited space on top of the sinks.
3. whiteboard - my oldest DS was able to come and go from Ocean Quest as he pleased and this helped us to inform him of where Mom and Dad were at any given time - also allowed him to leave us messages like "Mom - stopped by and you weren't here - gone to drink station to get a soda - be right back"
4. Sunblock
5. Hairdryer - some will tell you that the one in the stateroom is fine - and for some it is - but I found that I really needed my "tornado"
6. One of those pop-up type laundry hampers comes in handy when on a 7 day cruise - for the shorter cruises we just re-used plastic bags from the gift shops for our dirty laundry.
After that - things that aren't necessary - but we were happy to have (but we drove down to Port Canaveral which made this easy)
1. Beer
2. Wine
3. Bottled water
I can't think of anything else.