You could look at covered cat litter trays - I use one in a run. It is not watertight in heavy rain but does a good job otherwise. About £10 from local pet supplier.
I also have some green cat igloos in the runs. They are really good outside as they are one moulded piece of tough plastic, but expensive at around £50. Seen them in in P@H, Jolleyes, garden centres. They are usually up on the very top shelves at ceiling height round here. There are the rounded / arch top ones or the square-based pyramid roof ones.
If you want something as just a lid, some plastic stacking drawer type boxes have a nice big dip in the front. Turn it upside down and you have a nice bunny igloo. Wilkinsons / B&Q?