Mice are mammals, like humans, so their proteins tend to show more homology with human proteins than non-mammalian options. They are also actually more closely related to humans than cats or dogs due to the relatively recent (~80 million year) separation of lineages that led to modern rodents and primates.
Mice breed fairly quickly year-round and have relatively large litters. Laboratory strains are also quite tolerant of research lab conditions.
Mice are inexpensive to keep: they are small, require little food and care, etc.
People tend to have less opposition to research in mice due to their status as pest species.
Many genetic tools are available for mice, which makes them better models for the next generation of science, which means there are more genetic tools available for the generation that follows, etc. There are now many specialized mouse lines, some for particular diseases, some for particular techniques, etc.
Mice are well-studied, so experiments in mice can be easily compared to other experiments, and there are a lot of standards for protocols. Most experiments are done on inbred mice, which have the advantage of being nearly genetically identical to each other, so experiments done in different labs are done in (nearly) genetically identical subjects which can help with reproducibility and comparisons across studies (though there are also downsides to inbred animals; see below).
None of these factors make mice perfect model organisms. Many successful experiments and treatments in mice have not translated to human outcomes. Laboratory mice are typically highly inbred and may display characteristics thought to be "normal" that are actually the result of genetic drift, founder effects, etc. These can include preferences for alcohol, tendencies toward obesity, poor immune systems compared to wild mice, etc. Some get particular tumors early. Some have seizures. Some lab mice are blind at birth or shortly after. Others have hearing problems.
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