The correct answer is option 3: Colostrum.

Colostrum is the special type of milk produced by a lactating mother immediately after childbirth. It is rich in nutrients and antibodies that play a crucial role in the development of the immune response in newborn babies.

Option 1, breast milk produced after a month of childbirth, is not the correct answer. Breast milk composition changes over time, but colostrum is produced immediately after childbirth and is different from milk produced later on.

Option 2, transitional milk, is not correct either. Transitional milk is the milk produced between colostrum and mature milk. It is still beneficial for the baby, but it does not have the same immune-boosting properties as colostrum.

Option 4, mineralized milk, is incorrect as well. There is no special type of milk called mineralized milk that is specifically produced by lactating mothers for the immune development of newborn babies.

In conclusion, colostrum is the special type of milk produced by a lactating mother immediately after childbirth that is essential for the development of the immune response of a newborn baby.