In my opinion, cheese would be a great addition to Minecraft, which is made by placing milk in a cauldron and similar to copper, it would, over time, turn into cheese. It would turn yellow and then you can collect it, to get a block of cheese, which in a crafting table you could split into 9 cheese (items). You could either eat it for 2 hunger points each, or you could use it for a mouse trap, which I will explain soon.
The new mob which I think would fit with cheese, is the mouse, which spawns around the world, but can create holes and enter your house. They run away from players, but they go after your uncollected cheese from your cauldron! You could chase them, but in my opinion they should be too fast for that, so as mentioned above, mousetraps should be the solution, and also one more to be mentioned below. Mice should be weak, and passive too.
The mousetrap should be made from 3 wood planks, a tripwire hook and a piece of cheese. Mice will be attracted to it, and when they get to it... they should drop as an item, 'dead mouse'. Ways to get rid of the dead mouse are, either leaving it to despawn, like any other item, or, you could pick it up and feed it to your cat, or throw it away. Cats are the 'other' solution to roaming mice, and they can chase the mice and kill them, and eat them by themselves.
The final thing is a food that many players want - pizza! In my opinion, it should be made from 3 bread and 3 cheese, or if more is needed, crops/meat could be added.