Can you train your mouse to use a litter box? It is certainly worth a try, but you may not have success getting pet mice (or other small rodents) completely litter trained. Mice are actually rather clean animals, and often choose one part of their home to use as a bathroom area most of the time. They usually still either mark territory or have accidents out of the box, so the goal should be to get the mice to use a litter box most of the time. This will definitely make cage cleaning a lot easier and help keep odors to a minimum. Here are some tips to get your mice to start using a box, at least most of the time.
Choosing a Litter
Use a paper-based product (Yesterday's News, CareFresh, etc.) for the litter box. Clay litters tend to be dusty and there is always the risk of ingestion. Do not use clumping cat litters (too much risk of ingestion/obstruction or the litter getting stuck to your pet). Choose a different product than you use for the cage bedding, though. Yesterday's News is a product that works well (assuming you are not already using it as bedding!).
Litter Training Tips
Choose a small, shallow container for a litter box. A small plastic storage container might work as long as it not too easily tipped. Small corner litter pans are available and are a good choice as many animals prefer to use a corner. Keep an eye on the cage to see if there is a particular area that your mice have already picked as a "toilet" area. This is a good place to put the litter box.
Place some feces and urine soaked bedding in the litter box. This gives the mice the idea that this is the place to go. If you do happen to catch your mice using the box, give them a treat!
Clean the litter pan out regularly, but at least for the first while, always leave a little bit of waste in the pan so the mice recognize it as "the place to go." You can try scooping the litter if you wish, but it is probably easier to just use a shallow layer and dump and replace the litter (saving a bit of waste) every day or two.
Don't expect perfection. The idea is to contain most of the waste to make cleaning easier. Your mice will likely still want to mark other parts of their home with urine or feces, and will also have times when they don't bother going to the litter box. If you have your mice out of the cage for playtime, you can provide a second box outside of the cage, too.