P&O Oceana and Ventura are good for children and have many 7 day cruises throughout the year. Oceana 9 May, Ventura 22 May (half term so almost fully booked) Oceana 30 May.
Royal Caribbean are excellent for kids activities , with safe, well supervised clubs. Also a great variety of food from self service in the windjammer to the superb main dining rooms. Some RCI ships also have children's meal times (organised by the staff in the kid's club) which gives you a chance to eat in peace! I'm sure your 5 year old would meet new friends and have a ball. Evening entertainment is superb too - especially on the larger ships with an ice rink etc. We are taking our 6 year old grandson and his mum on Indy in July and can't wait! Check out Independence of the Seas which sails from Southampton to the Med and Voyager is good too - any Freedom or Voyager class have loads of activities, but the smaller ones are also good. Whichever you choose, watch for the age groups of the kids clubs as different lines have different ages and I know if your 5 year old is anything like our 6 year old, being in with the babies (e.g. age 2-5) will NOT be cool. I think RCI is 5-7 but check it out before booking. Whichever you choose - ENJOY!
Last July I went with family on Ocean Village 2, & the children were very well looked after & kept occupied & supervised. Also, the food, for the whole family--age range 5 to 68yrs-- was fantastic, both for variety & quality. The most amazing thing was that they never ran out of anything. The cabins we had were bigger & had far more storage than we'd had previously with RCL. I have been on The Independence too, & all the family are catered for fabulously, in every way.Enjoy your trip, whatever you choose.