- The Custom map solution.
You can use the oceanBlock map, witch consist in an infinite sea with a small island where you start will the minimum resources to make it interesting. Your cousin will have to make a new world, but if what he likes is the infinite ocean, that should do it. There are even other maps like this one on the same idea.
- The mod solution.
If you accept mods, you can use ex nihilo, a mod with the purpose of making every resource available as long as you have a sapling and a piece of dirt to plant it.
- The hard manual solution.
If she really wants to stay on this very map and to play vanilla, you can take inspiration on the stuff you get on the skyblock map. The four first items are here to make the play possible, the others to make it more interesting:
at least a piece of dirt. Only one will make things very hard
a bucket of lava, to make stone.
a source of food. It can be either a wheat or melon seed to be planted or a grass block on witch you will put bonemeal to get seeds. If you want him to suffer you can let him fish for all his food after he would have built a spider spawning zone, but this is nasty.
two villager eggs, so he will not be alone and have access to pretty much everything.
either 10 lava sources or obsidian, to go to the nether. Keep in mind that obsidian is hard to mine. The flint&steel can be traded with villagers.
You can add some other stuff like sugarcane or animal eggs if you want, but that is not mandatory.