Some studies (news article, another article, actual study) have shown that mice fecal matter could contribute to asthma, or otherwise cause allergies in young children.
In a study of eight cities, a team from Johns Hopkins University in
Baltimore, US, checked the contents of dust samples from the homes of
children with asthma.
The team found that 95% of those samples contained mouse allergens,
many involving feces or urine.
Other concerns include: mouse droppings can pose to be a choking hazard; mice can eat up your furniture, wiring, and insulation; they may clog plumbing etc. in your house, creating fire hazards; and much more.
Murine Typhus Fever or simply Typhus is spread by the bites of fleas
that have fed off of rodents. This infectious disease shows itself
with a rash and other symptoms include headache, backache, a high
fever that may last up to two weeks, nausea, vomiting, and abdominal
pain. Treatment for typhus fever includes antibiotics and over 98% of
people afflicted will recover. 10% to 60% of people who do not go on
antibiotic treatment will not survive
Living with a mice infestation is dangerous for human beings in general; it is only going to be more dangerous for a helpless child to be there.