Mice enjoy these not only for sleeping quarters, but also for larders to hide their stash. There are plastic cubby houses available, which are known as Hamster Igloos. The smaller types are suitable for mice and they split in half lengthwise and therefore easy to clean, although they are prone to being chewed and will eventually be destroyed.
There are cane cubby houses which are available from the NSWFRMC, these might be sold in pet stores as bird nests. These are fairly quickly chewed by mice, which will provide entertainment in itself. A down side to these types of houses is that they are difficult to clean.
Bird breeding boxes can also be a good cubby house for a mouse. These are fairly resistant to chewing but the main disadvantage is that they will be eventually stained by urine. The main advantage of this type of house is that they usually have a flip top lid for easy access to pets and are easily cleaned.
Another type of house that may be found in the bird section is coconut nests. There are two types of these - one is the husk which actually surround the coconut. This usually has a hole at one end and a segment which is removable to look inside. The second one is the hard coconut shell itself. This type usually have a single entry hole drilled into the coconut and the flesh has been removed. The husk type nest is prone to be chewed and may eventually become smelly. It is most suitable for rats. The coconut shell is quite small and is suitable for mice. This nest is quite resistant to chewing and fairly easy to clean.
There are also many types of homemade cubby houses available and these can often work out quite cheep which is good when you are replacing them regularly. The first most obvious homemade cubby house is a cardboard box. These can be boxes you find, tissue boxes (remove the plastic flaps), cereal boxes, boxes from packaging of toys, etc. None of these last very long. Generally they are chewed or get smelly very quickly.
Plastic screw-top jars can also make good cubby houses. They are easy to clean and can be attached to the side of a cage. Make two screw holes in the lid. With the top of the lid facing to the outside of the cage, screw the lid to a small piece of wood. The wood should be on the outside of the cage, the lid on the inside. Cut a hole in the bottom of the jar. Screw the jar to it's lid. The animals enter through the hole in the bottom of the jar. The jar needs only to be unscrewed and it can be cleaned and washed.